In this notebook, I will walk through solving a complete machine learning problem using hypothetical weather conditions inside a greenhouse dataset.

This is a supervised, regression machine learning task: given a set of data with targets (in this case the yield) included, we want to train a model that can learn to map the features (also known as the explanatory variables) to the target.

Supervised problem: we are given both the features and the target Regression problem: the target is a continous variable, in this case ranging from 0.38-1.07

The objective is to find a specific model for a greenhouse that can predict its yield with the accuracy of more than 92%.

Machine Learning Workflow

The structure of the machine learning project will be as follows:

  1. Data cleaning and formatting
  2. Exploratory data analysis
  3. Feature engineering and selection
  4. Establish a baseline and compare several machine learning models on performance metric (MAPE)
  5. Perform hyperparameter tuning on the best model to optimize it for the problem
  6. Evaluate the best model on the validation set and make prediction on the test set
  7. Interpret the model results to the extent possible
  8. Draw conclusions

Imports

I will use the standard data science and machine learning libraries: numpy, pandas, and scikit-learn. I also use matplotlib and seaborn for visualization.

In [422]:
# Pandas and numpy for data manipulation
import pandas as pd
import numpy as np

# No warnings about setting value on copy of slice
pd.options.mode.chained_assignment = None

# Display up to 60 columns of a dataframe
pd.set_option('display.max_columns', 60)

# Matplotlib visualization
import matplotlib.pyplot as plt
%matplotlib inline

# Set default font size
plt.rcParams['font.size'] = 24

# Internal ipython tool for setting figure size
from IPython.core.pylabtools import figsize

# Seaborn for visualization
import seaborn as sns
sns.set(font_scale = 2)

# Splitting data into training and testing
import random
from sklearn.model_selection import train_test_split

# Machine Learning Models
import lightgbm as lgb
from sklearn.linear_model import LinearRegression
from sklearn.ensemble import RandomForestRegressor, GradientBoostingRegressor
from sklearn.svm import SVR
from sklearn.neighbors import KNeighborsRegressor

# Hyperparameter tuning
from sklearn.model_selection import RandomizedSearchCV, GridSearchCV

# LIME for explaining predictions
import lime 
import lime.lime_tabular

Data Cleaning and Formatting

Load in the Data and Examine

In [423]:
data = pd.read_csv('dataset.csv')

# Display top of dataframe
data.head()
Out[423]:
week_num yield 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 ... 4311 4312 4313 4314 4315 4316 4317 4318 4319 4320 4321 4322 4323 4324 4325 4326 4327 4328 4329 4330 4331 4332 4333 4334 4335 4336 4337 4338 4339 4340
0 Week 48 FY15 0.57 0.765056 -0.356439 0.197052 0.795395 0 -0.001015 0.196354 0.081835 0.950103 -0.063364 -0.829207 -0.241069 1.456464 -0.330876 0.018767 0.260456 -0.067674 0.525733 -0.131478 -0.474496 -0.229206 -0.436584 -0.391205 -0.291944 -0.394279 0.015729 -0.661840 -0.221699 ... -1.3 -1.036887 -0.991674 -1.021653 -0.991608 -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994 -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988 -0.980759 -0.730907 -0.99136 -0.774115 -0.724674 -0.542333 -0.992039 -1.036513 -0.981902 -0.995691 -0.800299
1 Week 49 FY15 0.47 1.062557 -0.156156 -0.234454 0.950919 0 -0.170481 -0.057973 0.362018 0.590900 -0.030287 -0.808753 0.520046 -0.331018 1.098410 -0.036730 1.044749 -0.154249 0.656448 -0.159455 -0.758297 -0.190549 -0.332570 0.349147 0.341943 -0.310972 0.303210 -0.558677 -0.133704 ... -1.3 -1.036887 -0.991674 -1.021653 -0.991608 -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994 -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988 -0.980759 -0.730907 -0.99136 -0.774115 -0.724674 -0.542333 -0.992039 -1.036513 -0.981902 -0.995691 -0.800299
2 Week 50 FY15 0.43 0.961968 -0.276802 0.106828 1.075338 0 -0.004001 0.110006 0.201414 0.516113 -0.016122 -0.772992 0.347968 -0.331018 0.772261 0.225179 0.792992 -0.104244 0.553572 -0.175071 -0.563206 0.008438 -0.469626 0.413091 0.442866 -0.420743 0.414846 -0.595874 -0.101545 ... -1.3 -1.036887 -0.991674 -1.021653 -0.991608 -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994 -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988 -0.980759 -0.730907 -0.99136 -0.774115 -0.724674 -0.542333 -0.992039 -1.036513 -0.981902 -0.995691 -0.800299
3 Week 51 FY15 0.49 0.864231 -0.411954 0.174596 1.168652 0 0.040849 0.032461 0.311033 0.605321 -0.023123 -0.711317 -0.098511 -0.331018 -0.485911 -0.064177 0.610803 -0.110098 0.553419 -0.201746 0.084896 -0.078933 -0.542146 0.656361 0.552064 -0.478825 0.474788 -0.449464 -0.085681 ... -1.3 -1.036887 -0.991674 -1.021653 -0.991608 -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994 -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988 -0.980759 -0.730907 -0.99136 -0.774115 -0.724674 -0.542333 -0.992039 -1.036513 -0.981902 -0.995691 -0.800299
4 Week 52 FY15 0.57 -0.083462 -0.212342 0.110023 1.251599 0 0.021135 -0.133111 0.152851 0.944587 1.262242 -0.840133 1.034012 -0.331018 0.471113 -0.287214 0.642511 -0.140182 0.083886 -0.212156 -0.806272 -0.432690 -0.568484 0.453516 0.341360 -0.499920 0.375777 -0.458860 -0.107713 ... -1.3 -1.036887 -0.991674 -1.021653 -0.991608 -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994 -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988 -0.980759 -0.730907 -0.99136 -0.774115 -0.724674 -0.542333 -0.992039 -1.036513 -0.981902 -0.995691 -0.800299

5 rows × 4342 columns

Taking a look at the raw data, we can already see that the ranges of variables' values are not very different. Although all I know from the problem statement is that we want to predict the number in the yield column that has a masure unit kg/m2, and the rest of the varibles are just numbered, I can make an educated guess and say that the variables have been standardized/normalized due to different unit measures and value ranges.

Data Types and Missing Values

Further, I will take a look at the data types to make sure they make sense(have the coressponding data type assumed correct)

In [424]:
# See the column data types and non-missing values
data.info()
<class 'pandas.core.frame.DataFrame'>
RangeIndex: 131 entries, 0 to 130
Columns: 4342 entries, week_num to 4340
dtypes: float64(4327), int64(14), object(1)
memory usage: 4.3+ MB

Most of the variables seem to be numeric, except the week number column beeing object type (string). To be later checked why 14 variables are innt64 and not float type.

In [425]:
# Statistics for each column
data.describe()
Out[425]:
yield 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 ... 4311 4312 4313 4314 4315 4316 4317 4318 4319 4320 4321 4322 4323 4324 4325 4326 4327 4328 4329 4330 4331 4332 4333 4334 4335 4336 4337 4338 4339 4340
count 115.000000 131.000000 131.000000 131.000000 131.000000 131.0 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 ... 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000
mean 0.675391 0.012175 -0.011646 0.032254 0.026639 0.0 -0.061272 -0.039280 -0.022693 0.009954 -0.006165 0.010942 -0.093526 -0.070380 -0.006834 -0.003581 0.007788 0.008028 -0.003892 -0.043079 0.019994 0.028660 -0.019106 0.000588 -0.000803 -0.025285 0.000811 -0.017509 0.005951 0.016551 ... 0.019847 -0.018212 0.015140 -0.012253 0.015139 0.013703 -0.013526 0.015146 0.013796 -0.017083 0.019840 -0.013002 0.019632 -0.017392 0.014745 0.015139 -0.010341 -0.023997 0.014198 0.014973 0.011159 0.015135 -0.000732 0.011064 0.001800 0.015146 -0.016099 0.014991 -0.016266 -0.018191
std 0.173095 0.992832 0.998961 0.974747 0.987683 0.0 0.671533 0.938874 0.968220 1.000330 1.008214 1.007417 0.574728 0.820738 1.009729 1.010899 1.009461 1.009276 1.000959 0.940680 0.998142 0.945582 0.973479 1.010564 1.010149 0.953264 1.010806 0.983389 1.009957 1.002341 ... 0.998339 1.000419 1.003851 1.006448 1.003852 1.005233 1.005352 1.003845 1.005148 1.001739 0.998349 1.005716 0.998624 1.001403 1.004245 1.003852 1.007746 0.992102 1.004773 1.004018 1.007339 1.003856 1.006302 1.007410 1.009994 1.003845 1.002811 1.004001 1.002566 1.000447
min 0.380000 -2.276594 -0.600320 -1.344745 -1.765569 0.0 -0.428650 -7.752403 -8.845893 -3.391754 -0.186073 -0.885389 -1.071456 -0.331018 -1.373330 -5.433454 -2.357776 -0.415026 -3.336172 -0.240134 -1.307903 -7.622581 -0.568484 -10.386934 -9.415406 -0.499920 -8.203347 -0.739742 -1.229447 -1.150176 ... -1.300000 -1.036887 -0.991674 -1.021653 -0.991608 -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994 -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988 -0.980759 -0.730907 -0.991360 -0.774115 -0.724674 -0.542333 -0.992039 -1.036513 -0.981902 -0.995691 -0.800299
25% 0.525000 -0.726656 -0.445459 -0.786582 -0.733926 0.0 -0.277096 -0.212699 -0.212299 -0.212930 -0.125539 -0.753347 -0.460810 -0.331018 -0.739639 -0.160585 -0.418765 -0.286940 -0.653723 -0.220940 -0.637095 -0.097085 -0.544300 -0.175117 -0.156850 -0.474049 -0.070744 -0.549544 -0.202120 -0.805935 ... -1.300000 -1.036887 -0.991674 -1.021653 -0.991608 -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994 -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988 -0.980759 -0.730907 -0.991360 -0.774115 -0.724674 -0.542333 -0.992039 -1.036513 -0.981902 -0.995691 -0.800299
50% 0.650000 0.131757 -0.351201 -0.017179 0.162930 0.0 -0.149212 0.104851 -0.024938 0.185155 -0.110266 -0.408105 -0.229996 -0.331018 -0.285495 -0.025708 0.314754 -0.104244 0.258382 -0.201746 -0.243295 0.129581 -0.334035 0.083081 0.126280 -0.310972 0.096966 -0.317995 -0.174289 -0.164899 ... 0.588668 0.894109 0.936147 0.457846 0.862280 -0.260324 0.470975 0.948911 -0.703088 0.537219 0.617484 -0.121098 0.582681 -0.811748 0.492216 0.934064 -0.752612 0.357856 -0.353018 0.534925 -0.730907 0.927847 -0.178247 -0.202833 -0.333432 0.927506 0.839520 0.742849 0.401413 -0.800299
75% 0.820000 0.868233 -0.249513 0.689194 0.930182 0.0 0.022078 0.409939 0.128273 0.416063 -0.081012 0.559834 0.113267 -0.331018 0.653403 0.160909 0.789570 0.104917 0.736178 -0.179883 0.497446 0.415257 0.155760 0.368455 0.348405 0.045178 0.338238 0.054899 -0.133377 0.484090 ... 0.842512 0.941332 0.990198 1.010513 1.013364 0.861445 1.009165 1.001910 1.148301 0.961256 0.861348 1.022361 0.879483 1.184753 0.932457 0.986209 0.903354 0.870551 0.958373 1.049698 1.022801 0.985124 0.253157 0.328903 0.009021 1.000359 0.960832 0.950500 0.838910 1.095020
max 1.070000 1.711989 2.985258 2.743958 1.251599 0.0 7.262550 1.142294 3.174728 7.053521 11.358507 3.373740 3.117461 4.307555 2.872462 6.954217 1.312071 11.140575 2.087265 7.907794 3.220929 0.873747 4.481160 1.102200 1.056587 4.234904 0.828886 3.591991 7.193852 3.744339 ... 0.995714 1.200532 1.182414 1.316671 1.102186 2.091451 1.323212 1.125495 1.559355 1.204925 0.968651 1.527287 1.077952 1.405106 1.871936 1.188678 2.651805 1.949017 1.764962 1.386134 1.936779 1.223061 2.690308 3.203145 3.669438 1.187765 1.114567 1.424802 1.683484 1.713457

8 rows × 4341 columns

Missing Values

Now that we have the correct column datatypes, we can start analysis by looking at the percentage of missing values in each column.

Following is a function that calculates the number of missing values and the percentage of the total values that are missing for each column.

In [426]:
# Function to calculate missing values by column
def missing_values_table(df):
        # Total missing values
        mis_val = df.isnull().sum()
        
        # Percentage of missing values
        mis_val_percent = 100 * df.isnull().sum() / len(df)
        
        # Make a table with the results
        mis_val_table = pd.concat([mis_val, mis_val_percent], axis=1)
        
        # Rename the columns
        mis_val_table_ren_columns = mis_val_table.rename(
        columns = {0 : 'Missing Values', 1 : '% of Total Values'})
        
        # Sort the table by percentage of missing descending
        mis_val_table_ren_columns = mis_val_table_ren_columns[
            mis_val_table_ren_columns.iloc[:,1] != 0].sort_values(
        '% of Total Values', ascending=False).round(1)
        
        # Print some summary information
        print ("Your selected dataframe has " + str(df.shape[1]) + " columns.\n"      
            "There are " + str(mis_val_table_ren_columns.shape[0]) +
              " columns that have missing values.")
        
        # Return the dataframe with missing information
        return mis_val_table_ren_columns
In [427]:
missing_values_table(data)
Your selected dataframe has 4342 columns.
There are 1 columns that have missing values.
Out[427]:
Missing Values % of Total Values
yield 16 12.2

Although we want to be careful to not discard information and should be careful when dropping columns, if a column has a high percentage of missing values, then it probably will not be of much use.

What columns to retain may be a little arbitrary here is a discussion, but for this project, if has been the case- I would have removed any columns with more than 50% missing values.

Here the only missing values are those in yield column, which correspond to the future trainig test split, so is as it is intentionally.

In [428]:
# Get the columns with > 50% missing (generic)
missing_df = missing_values_table(data);
missing_columns = list(missing_df[missing_df['% of Total Values'] > 50].index)
print('We will remove %d columns.' % len(missing_columns))
Your selected dataframe has 4342 columns.
There are 1 columns that have missing values.
We will remove 0 columns.
In [429]:
#Select the int64 type variables for inspection
data.select_dtypes(include=['int64']).head(10)
Out[429]:
5 315 625 935 1245 1555 1865 2175 2485 2795 3105 3415 3725 4035
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0 0 0 0 0 0 0
4 0 0 0 0 0 0 0 0 0 0 0 0 0 0
5 0 0 0 0 0 0 0 0 0 0 0 0 0 0
6 0 0 0 0 0 0 0 0 0 0 0 0 0 0
7 0 0 0 0 0 0 0 0 0 0 0 0 0 0
8 0 0 0 0 0 0 0 0 0 0 0 0 0 0
9 0 0 0 0 0 0 0 0 0 0 0 0 0 0

It seems that these variables have constant value of zero, hence the int64 data type. Not sure if it's normal, as I am not aware of the columns significance.

Removing Outliers

When we remove outliers, we want to be careful that we are not throwing away measurements just because they look strange. They may be the result of actual phenomenon that we should further investigate. When removing outliers, I try to be as conservative as possible, using the definition of an extreme outlier:

  • On the low end, an extreme outlier is below $\text{First Quartile} -1.5 * \text{Interquartile Range}$
  • On the high end, an extreme outlier is above $\text{Third Quartile} + 1.5 * \text{Interquartile Range}$

I will define a function that updates all the outliers with the appropriate value in all the columns in the dataset.

In [430]:
#function which takes input a dataframe(x) that can contain both numeric and categorical columns
def fixoutliers(x):
    #Get all the column name from the input dataframe x
       xColumnNames=x
       #keep the labels column as we do't want to alter it 
       y = xColumnNames['yield']
       xColumnNames.drop('yield', axis=1)
       print(xColumnNames)
#go throug all collumns ans find whether the values are whiting the setted range:

       for j in xColumnNames:
            try:
              print ("colnames ",j)
              xy=x[j]    
              mydata=pd.DataFrame()
        #make an empty list to add the uppdated values 
              updated=[]
              Q1,Q3=np.percentile(xy,[25,75])
              IQR=Q3-Q1
              minimum=Q1-1.5*IQR #maximum value
              maximum=Q3+1.5*IQR #minimum value
              for i in xy:   #alter columns if they are not in within the range
                 if(i>maximum):
                   print("Entering maxim")
                   i=maximum
                   updated.append(i)
                 elif(i<minimum):
                  print("enterinf minimum")
                  i=minimum
                  updated.append(i)
                 else:
                  updated.append(i)
              x[j]=updated #update the dataframe with the updated values
            except:
              continue
            
       x['yield']=y     #add back the labels column
       return x
In [431]:
#fixing outliers
new_data=fixoutliers(data)
         week_num  yield         1         2         3         4  5         6  \
0    Week 48 FY15   0.57  0.765056 -0.356439  0.197052  0.795395  0 -0.001015   
1    Week 49 FY15   0.47  1.062557 -0.156156 -0.234454  0.950919  0 -0.170481   
2    Week 50 FY15   0.43  0.961968 -0.276802  0.106828  1.075338  0 -0.004001   
3    Week 51 FY15   0.49  0.864231 -0.411954  0.174596  1.168652  0  0.040849   
4    Week 52 FY15   0.57 -0.083462 -0.212342  0.110023  1.251599  0  0.021135   
5    Week 53 FY15   0.63 -0.456798 -0.264498  0.322492  1.251599  0  0.065378   
6     Week 1 FY16   0.57 -0.851854 -0.354455 -0.017179  1.251599  0 -0.038447   
7     Week 2 FY16   0.56 -0.596559 -0.225321 -0.090215  1.199757  0 -0.040010   
8     Week 3 FY16   0.56 -0.706136 -0.512842  0.145560  1.106443  0  0.007287   
9     Week 4 FY16   0.63 -0.941584 -0.272370  0.069204  0.982023  0 -0.056107   
10    Week 5 FY16   0.62 -0.820188 -0.140776  0.021004  0.836868  0 -0.047715   
11    Week 6 FY16   0.68 -0.947957 -0.252931  0.328052  0.660607  0 -0.001150   
12    Week 7 FY16   0.60 -0.072560 -0.389566  0.905665  0.463610  0  0.128607   
13    Week 8 FY16   0.77  0.641590 -0.360295  0.874913  0.256244  0  0.093180   
14    Week 9 FY16   0.84  0.962713 -0.344281  0.981958  0.038511  0  0.151755   
15   Week 10 FY16   0.78  0.701348 -0.426749  1.541777 -0.179223  0  0.261833   
16   Week 11 FY16   0.80  1.239605 -0.212466  1.099610 -0.428061  0 -0.026428   
17   Week 12 FY16   0.79  0.614912 -0.336554  0.318028 -0.656163  0  0.026058   
18   Week 13 FY16   0.97 -0.005624 -0.500801  0.751983 -0.023699  0  0.175924   
19   Week 14 FY16   0.82  0.042397 -0.535337  0.627734 -0.251801  0  0.116883   
20   Week 15 FY16   0.92 -0.590055 -0.417868  1.059106 -0.479903  0  0.167668   
21   Week 16 FY16   0.86 -0.376581 -0.253219  1.460796 -0.708005  0  0.248349   
22   Week 17 FY16   0.82  0.222713 -0.180255  0.476057 -0.915370  0 -0.033988   
23   Week 18 FY16   0.96 -1.232107 -0.468408  1.987128 -1.112367  0  0.400167   
24   Week 19 FY16   0.85 -1.551303 -0.180924  2.717746 -1.288628  0 -0.041838   
25   Week 20 FY16   0.87 -0.955122 -0.386764  0.578831 -1.444152  0  0.046791   
26   Week 21 FY16   0.97 -0.564804 -0.361657  0.607262 -1.568571  0 -0.144174   
27   Week 22 FY16   0.94 -0.517209 -0.290588  0.132272 -1.672254  0 -0.273346   
28   Week 23 FY16   0.69 -1.558668 -0.556270  2.055333 -1.724095  0  0.038877   
29   Week 24 FY16   0.95 -0.979617 -0.600035  0.185035 -1.765569  0 -0.076448   
..            ...    ...       ...       ...       ...       ... ..       ...   
101  Week 46 FY18   0.44 -1.997424 -0.464488 -0.686207  0.525819  0 -0.228737   
102  Week 47 FY18   0.41 -0.156663 -0.252901 -1.124501  0.691712  0 -0.368212   
103  Week 48 FY18   0.51  0.044784 -0.312652 -1.066308  0.867972  0 -0.284861   
104  Week 49 FY18   0.47  0.236870 -0.207224 -1.022153  1.002760  0 -0.292725   
105  Week 50 FY18   0.38  1.190334 -0.281346 -0.796685  1.137548  0 -0.241557   
106  Week 51 FY18   0.48  1.141279 -0.289596 -0.788617  1.210125  0 -0.316182   
107  Week 52 FY18   0.46  1.227195 -0.582593 -0.647916  1.251599  0 -0.153174   
108   Week 1 FY19   0.53  1.341313 -0.416086 -0.663768  1.251599  0 -0.154038   
109   Week 2 FY19   0.58  1.337377  2.623997 -0.630308  1.241230  0 -0.174479   
110   Week 3 FY19   0.54  1.538685  2.754314 -0.639562  1.158284  0 -0.177734   
111   Week 4 FY19   0.49  1.680356  2.862073 -0.593140  1.064970  0 -0.134579   
112   Week 5 FY19   0.52  1.530294  2.778689 -0.540858  0.909446  0 -0.141322   
113   Week 6 FY19   0.56  1.477546  2.778379 -0.426826  0.753921  0 -0.161914   
114   Week 7 FY19   0.60  0.991019  2.538383 -0.157213  0.567293  0 -0.158918   
115   Week 8 FY19    NaN  0.956316  2.628104  0.214427  0.370295  0 -0.009046   
116   Week 9 FY19    NaN  0.814405  2.627568  0.522202  0.152562  0  0.051672   
117  Week 10 FY19    NaN  1.275518  2.985258  0.696234 -0.075540  0 -0.042313   
118  Week 11 FY19    NaN  1.470095  2.911881  0.029781 -0.293274  0 -0.007514   
119  Week 12 FY19    NaN  0.915698  2.754974  0.721501 -0.531744  0  0.011705   
120  Week 13 FY19    NaN  0.791334  2.691973  0.784963 -0.759846  0  0.077915   
121  Week 14 FY19    NaN  1.007394  2.757787  0.530768 -0.137750  0 -0.024650   
122  Week 15 FY19    NaN  0.872236  2.858886  1.159055 -0.365852  0  0.225152   
123  Week 16 FY19    NaN  0.327093  2.477116  1.088888 -0.583586  0  0.023021   
124  Week 17 FY19    NaN  0.041694 -0.302001  1.475804 -0.801319  0  0.119715   
125  Week 18 FY19    NaN  1.711989 -0.532453  0.091467 -0.998316  0 -0.171679   
126  Week 19 FY19    NaN  0.913389 -0.322707  0.881106 -1.195314  0 -0.116319   
127  Week 20 FY19    NaN  0.763270 -0.323498  1.335526 -1.361206  0 -0.006336   
128  Week 21 FY19    NaN  0.931748 -0.459942  0.445836 -1.495993  0 -0.235421   
129  Week 22 FY19    NaN  0.175665 -0.378380 -0.213004 -1.620413  0 -0.237201   
130  Week 23 FY19    NaN  0.131757 -0.453177 -1.164508 -1.703359  0 -0.417608   

            7         8         9        10        11        12        13  \
0    0.196354  0.081835  0.950103 -0.063364 -0.829207 -0.241069  1.456464   
1   -0.057973  0.362018  0.590900 -0.030287 -0.808753  0.520046 -0.331018   
2    0.110006  0.201414  0.516113 -0.016122 -0.772992  0.347968 -0.331018   
3    0.032461  0.311033  0.605321 -0.023123 -0.711317 -0.098511 -0.331018   
4   -0.133111  0.152851  0.944587  1.262242 -0.840133  1.034012 -0.331018   
5    0.026284  0.204981  0.886062 -0.025361 -0.722637  0.455690 -0.331018   
6    0.111465  0.005072  0.686089 -0.060036 -0.835025 -0.616237  1.202706   
7   -0.164893 -0.233994  0.717350 -0.065681 -0.727746  0.953961  4.307555   
8    0.076960 -0.011042  0.721449 -0.094207 -0.624133 -0.602433 -0.331018   
9   -0.045328 -0.173382  0.524688 -0.086238 -0.772276 -0.018838 -0.331018   
10  -0.081584 -0.164401  0.261906 -0.089863 -0.597862  0.782933 -0.331018   
11  -0.104150 -0.253139  0.258368 -0.074057 -0.713154 -0.241594 -0.331018   
12   0.466731 -0.232460  0.312260 -0.092880 -0.549029 -0.489206 -0.331018   
13   0.409173 -0.109481  0.463014 -0.081604 -0.530731 -0.555285 -0.331018   
14   0.538683 -0.178372  0.222855 -0.105321 -0.527097 -0.311497 -0.331018   
15   0.992504  0.011226  0.448058 -0.114546  0.390271 -0.342761 -0.331018   
16   1.073600 -0.165750  0.726581 -0.120892 -0.219126  0.777743 -0.331018   
17   0.892368 -0.162184  0.675794 -0.125995 -0.033747  0.507335 -0.331018   
18   0.662722 -0.006607 -0.062188 -0.120090  0.659446 -1.026724 -0.331018   
19   0.451581 -0.277716  0.320034 -0.121037  0.464684 -0.576905 -0.331018   
20   0.590672 -0.252714  0.389844 -0.121558  1.168154 -0.223147 -0.331018   
21   0.632910 -0.123507 -0.092268 -0.110133  1.705222  0.734650 -0.331018   
22  -0.108333 -0.686366  0.335339 -0.124389  0.554411  0.409109  3.759834   
23   0.124086  0.107373 -0.152288 -0.104250  2.697795 -0.102440 -0.331018   
24  -0.011601  0.107355 -2.436139 -0.076029  0.604772  0.594277 -0.331018   
25   0.369318 -0.159707 -0.653411 -0.135414  1.403858  0.169671 -0.331018   
26   0.568690 -0.456096 -1.174938 -0.159057  0.366059  0.048292 -0.331018   
27   0.655865 -0.545093 -0.569883 -0.162297  0.069151 -0.777558 -0.331018   
28   0.712620  0.226084 -0.039805 -0.077401  1.270997 -0.675592 -0.331018   
29   0.400006 -0.198680  0.460085 -0.146174  0.912050 -0.749781 -0.331018   
..        ...       ...       ...       ...       ...       ...       ...   
101 -0.017291  0.096452  0.579671 -0.082677 -0.204346  0.045035 -0.331018   
102 -2.481125 -0.204150 -0.157867 -0.108390 -0.311422  0.347905 -0.331018   
103 -0.140455 -0.082871 -0.566605 -0.110273 -0.390021  0.520277 -0.331018   
104 -0.475124 -0.268014 -0.250756 -0.107079 -0.463387 -0.361166  0.046587   
105 -0.468242 -0.382918 -0.177491 -0.110266 -0.121495 -0.099771 -0.331018   
106 -0.334963 -0.213907  0.070532 -0.113140 -0.374715  0.561522 -0.331018   
107 -0.266220 -0.154700  0.072090 -0.115721 -0.490740 -0.759888 -0.331018   
108 -0.131141 -0.017380  0.083461 -0.126247 -0.592180  0.578183  0.888491   
109  0.117398 -0.064133  0.124783 -0.124601 -0.848698  1.666709  1.853431   
110  0.155186 -0.024938  0.118389 -0.115340 -0.880404 -0.607041 -0.331018   
111  0.106772 -0.344001  0.100375 -0.117429 -0.880760 -0.613190 -0.331018   
112  0.238690 -0.123525  0.137298 -0.135322 -0.883808 -0.665871 -0.331018   
113  0.015439 -0.558749 -0.104596 -0.130003 -0.882902 -0.650211 -0.331018   
114  0.245158 -0.068716 -0.546196 -0.124727 -0.882003 -0.634677 -0.331018   
115  0.194069  0.303124 -0.116936 -0.117847 -0.880831 -0.614423 -0.331018   
116  0.149132  0.247169 -0.424332 -0.125186 -0.882081 -0.636029 -0.331018   
117  0.499218  0.039998  0.079955 -0.142627 -0.856631  1.277495  1.534318   
118  0.410705  0.007216  0.060708 -0.136340 -0.829214  3.117461  3.263501   
119  0.851079  0.267589 -0.348971 -0.143305 -0.885167 -0.689375 -0.331018   
120  0.743940  0.437524 -0.463685 -0.144606 -0.885389 -0.693206 -0.331018   
121  0.386843  0.046468 -0.011987 -0.142603 -0.885048 -0.687309 -0.331018   
122  0.161971  0.126943 -0.162100 -0.099338 -0.877678 -0.559928 -0.331018   
123 -1.007487  0.275830 -2.458712 -0.125067 -0.882061 -0.635679 -0.331018   
124 -1.899480  0.123099 -2.469888 -0.118949  1.088955  0.327062 -0.331018   
125 -1.222997  1.483795 -0.056542 -0.167613 -0.749727 -1.071456 -0.331018   
126 -0.250421  2.898830  0.387723 -0.147924  3.325954  0.383476 -0.331018   
127 -1.619412  0.692612  0.420410 -0.135193  0.966066  0.081363 -0.331018   
128 -1.861440  0.586171  0.404269 -0.186073 -0.732222 -0.298029 -0.331018   
129  0.849352  3.174728  0.327388 -0.149763 -0.732222 -0.121055 -0.331018   
130  0.550550  1.706273 -0.248368 -0.070427 -0.767232 -0.405835  3.517837   

           14        15        16         17        18        19        20  \
0   -0.330876  0.018767  0.260456  -0.067674  0.525733 -0.131478 -0.474496   
1    1.098410 -0.036730  1.044749  -0.154249  0.656448 -0.159455 -0.758297   
2    0.772261  0.225179  0.792992  -0.104244  0.553572 -0.175071 -0.563206   
3   -0.485911 -0.064177  0.610803  -0.110098  0.553419 -0.201746  0.084896   
4    0.471113 -0.287214  0.642511  -0.140182  0.083886 -0.212156 -0.806272   
5    0.286275 -0.069307  0.858496  -0.081506 -1.018462 -0.178324 -1.069046   
6   -0.423202  0.001739  0.915290  -0.113654 -1.013593 -0.155552 -1.284865   
7    1.359882 -0.209458  0.966760  -0.163612 -0.869547 -0.155552 -1.270213   
8    0.415522  0.012373  0.968453  -0.117103 -0.992087 -0.214108 -1.306961   
9   -0.463773 -0.138527  0.875731  -0.090379 -1.569212 -0.167914 -1.286444   
10   0.660597 -0.216902  0.843484  -0.117519 -1.099184 -0.190035 -1.292556   
11   2.396268  0.046142  0.827524  -0.068819 -1.312313 -0.193939 -0.244044   
12  -0.642629  0.071649  0.806780   0.099884  0.289107  4.238547  0.080191   
13  -0.539470  0.006064  0.568630   0.128270  0.896273 -0.183529 -0.243295   
14   1.356695  0.164652  0.767695   0.136826  1.323942 -0.176372  0.758757   
15  -0.239914  0.219359  0.478210   0.215871  0.427432 -0.203048 -0.181267   
16  -0.611204  0.090230  0.333046   0.298392  1.002711 -0.237531 -0.732926   
17  -0.529565  0.199873 -0.168242  -0.010124  0.969138 -0.170516 -0.064568   
18   0.391084  0.214904 -0.286220   0.007500  0.277622 -0.179490 -0.013608   
19   0.580254  0.114386 -0.284111   0.024389  0.059211 -0.207689  0.015035   
20  -0.532520  0.112633 -0.736536   0.126025 -0.669890 -0.197192 -0.164284   
21   1.165021  0.076377 -0.220387   0.225819 -0.253067 -0.223868 -0.427436   
22   1.103284  0.151116 -0.090487   0.014762  0.768613 -0.195240 -0.851257   
23   2.872462  0.170328 -0.430248   0.315406 -1.425076 -0.224518 -0.411117   
24   0.646209  0.069120 -0.497872   0.396556 -1.350444 -0.229073 -0.755139   
25   0.585800  0.107488 -0.407282   0.135924 -1.218127 -0.218663 -0.706833   
26  -1.061113  0.628819 -2.194631   0.258095  0.363896 -0.165311 -0.723376   
27  -1.311875 -5.433454 -2.357776   0.087742  0.503261 -0.203048 -0.718167   
28  -0.611304 -3.855242 -2.232125   0.328612 -1.779765 -0.199795 -0.733822   
29  -1.162108  0.206254 -1.993135  -0.049024 -0.637555 -0.201746 -0.588583   
..        ...       ...       ...        ...       ...       ...       ...   
101  0.745304 -0.059478  0.518076  -0.233042 -1.591581 -0.201096  0.585690   
102 -0.307889 -0.227653  0.681104  -0.352122 -0.409276 -0.238832 -0.405324   
103 -1.232009 -0.178332  0.613089  -0.347178 -0.239956 -0.190686 -0.199957   
104 -1.268835 -0.041314  0.633217  -0.353971  0.069782 -0.188083  0.925029   
105  1.301768 -0.324806  0.880085  -0.312220  1.182064 -0.202397  0.719138   
106 -1.364689 -0.227464  0.743433  -0.314560  0.607593 -0.187433  0.666853   
107 -0.645425 -0.231401  0.681280  -0.287614  0.618124 -0.159455  2.949712   
108 -0.896454 -0.258503  0.681271  -0.295416  0.825442 -0.205000  0.070430   
109 -0.472228 -0.117547  0.698322  -0.233379  0.746016 -0.204349  0.088139   
110 -0.826189 -0.215120  0.800300  -0.286265  1.298502 -0.191987  1.826623   
111 -0.985353 -0.121513  1.088267  -0.267869  1.633421 -0.203698  1.381248   
112 -1.324062 -0.113466  0.700930  -0.255413  1.024095 -0.150997  1.695526   
113 -0.988694 -0.032519  0.777016  -0.222856  1.410345 -0.150347  0.693930   
114 -0.870961 -0.207935  0.575570  -0.127631  0.930368  7.907794  3.220929   
115 -0.627673  0.108293  0.264360  -0.080899  0.626586 -0.224518  2.184864   
116 -0.005022  0.096782  0.265974  -0.001924  0.557648 -0.223868  2.945782   
117 -0.606199  0.204946  0.143831  -0.039035  0.751784 -0.171167  0.549033   
118  0.177966 -0.043383  0.079811  -0.122357  1.199627 -0.157504  0.216209   
119 -0.164180  0.295305 -0.001509   0.060160  0.651803 -0.207602 -0.544780   
120  0.066180  0.267254  0.075234   0.079044  0.815375 -0.229913 -0.560245   
121  0.614225  0.045568  0.330874   0.109950  0.714918 -0.225820 -0.570152   
122  2.852251 -0.111943  0.065266   0.129880  0.808389 -0.237531 -0.604914   
123  1.383475 -0.100993  0.032956   0.172748  0.755833 -0.236230 -0.694673   
124  1.381297  0.437410 -0.532210   0.664551  0.699102 -0.204224 -1.307903   
125 -0.718731  0.648693 -0.233320   0.117463  1.530608 -0.227076 -0.876869   
126  1.584555  6.954217 -0.273943   0.211800  1.109634 -0.237531 -0.669515   
127 -0.078226  3.961915 -0.371109   0.242729  1.078310 -0.237522 -0.549518   
128 -1.370838 -0.886246 -1.119586  11.140575  2.087265 -0.215892 -0.960731   
129 -0.456891  1.568866 -0.849943   0.040423  0.586912 -0.205530 -0.582293   
130  1.114148  0.447900 -0.190002  -0.280591  0.327010 -0.238828 -0.617789   

           21        22        23        24        25        26        27  \
0   -0.229206 -0.436584 -0.391205 -0.291944 -0.394279  0.015729 -0.661840   
1   -0.190549 -0.332570  0.349147  0.341943 -0.310972  0.303210 -0.558677   
2    0.008438 -0.469626  0.413091  0.442866 -0.420743  0.414846 -0.595874   
3   -0.078933 -0.542146  0.656361  0.552064 -0.478825  0.474788 -0.449464   
4   -0.432690 -0.568484  0.453516  0.341360 -0.499920  0.375777 -0.458860   
5    0.003995 -0.568484  0.565111  0.462719 -0.499920  0.455587 -0.471250   
6    0.017631 -0.140339  0.159893  0.098676 -0.157010  0.238887 -0.695173   
7   -0.385160  0.107857 -0.172452 -0.094228  0.041776 -0.000249 -0.638167   
8   -0.207383  0.057243  0.083081  0.030260  0.001238  0.102722 -0.720894   
9    0.110535  0.136234 -0.217425 -0.112903  0.283213 -0.073238 -0.599283   
10  -0.003760  0.165570 -0.152473 -0.067827  0.287080 -0.022925 -0.662557   
11   0.129581  0.462212 -0.149125 -0.060314  0.325586 -0.031248 -0.694230   
12  -0.046415 -0.104676 -0.103934 -0.152117 -0.128446 -0.030776 -0.683894   
13   0.427678 -0.568484 -0.002733  0.000292 -0.499920  0.045767 -0.555230   
14   0.286190 -0.568484 -0.069337 -0.341459 -0.499920 -0.177391 -0.559844   
15   0.765452 -0.489739  0.721620  0.645725 -0.436851  0.565681 -0.480762   
16   0.504792 -0.546406 -0.089311 -0.010750 -0.482238 -0.097800 -0.258370   
17   0.607234 -0.568484 -0.003053  0.058704 -0.499920  0.033373 -0.283569   
18   0.643958 -0.568484  0.310421  0.279676 -0.499920  0.300648 -0.478274   
19   0.330150 -0.568484 -0.268743 -0.302541 -0.499920 -0.147934 -0.237352   
20  -0.041635 -0.516771 -0.017690 -0.058736 -0.458386  0.050702 -0.068272   
21  -0.028204 -0.461145  0.191189  0.334422 -0.413950  0.236231  0.219275   
22  -0.063212  0.126826 -0.863916 -0.726254  0.184865 -0.435174 -0.317995   
23  -0.179023 -0.568484  0.518470  0.410860 -0.499920  0.455340 -0.050993   
24  -0.854044 -0.568484 -0.133236  0.126280 -0.499920 -0.004816  0.495217   
25   0.419148 -0.544808 -0.083650 -0.081212 -0.480957 -0.001743  0.314583   
26   0.287860 -0.547127 -0.699934 -0.425129 -0.459374 -0.398808  0.118886   
27   0.055985 -0.496526 -0.942506 -0.562969 -0.442288 -0.627823  0.552068   
28   0.141815 -0.489956 -0.104117 -0.035700 -0.437026 -0.056665  0.540991   
29   0.037330 -0.568484 -0.555050 -0.394710 -0.499920 -0.235796  0.396355   
..        ...       ...       ...       ...       ...       ...       ...   
101  0.187974 -0.072408  0.708166  0.562456 -0.105944  0.508133 -0.485851   
102 -4.582335 -0.010985  0.377748  0.352635 -0.088901  0.370467 -0.555530   
103  0.360218  0.257467  0.470499  0.400314 -0.039999  0.255355 -0.639403   
104  0.243969  0.430726  0.173336  0.257725  0.451538  0.107486 -0.381452   
105  0.190549  0.765382  0.228255  0.249620  0.036343  0.048449 -0.541072   
106  0.107101  0.453365  0.200321  0.213517 -0.002526  0.124509 -0.609261   
107  0.143889  0.707044  0.220298  0.233150  0.344315  0.185072 -0.540370   
108  0.278771  0.218694  0.308675  0.333742  0.134948  0.156100 -0.463140   
109  0.310102 -0.094483  0.135590  0.225277 -0.350665  0.067770 -0.546253   
110  0.203643  0.626478  0.069186  0.149552 -0.061539  0.016429 -0.429644   
111 -0.067302  0.427408 -0.567768 -0.530252  0.256933 -0.224993 -0.698875   
112  0.362053 -0.227161  0.209911  0.261317 -0.220218  0.145948 -0.739742   
113 -0.023610 -0.334035 -0.419251 -0.486912 -0.308793 -0.297192 -0.613146   
114  0.376248 -0.126826  0.283245  0.288654 -0.142103  0.209860 -0.382870   
115  0.448386 -0.231507  0.546950  0.523869 -0.220795  0.429462 -0.086437   
116  0.419390 -0.140384  0.532870  0.463003 -0.154742  0.400858 -0.119698   
117  0.642097 -0.201530  0.163459  0.150771 -0.202363  0.096966 -0.073168   
118  0.570900 -0.040500  0.301346  0.212235 -0.071163  0.171788 -0.207223   
119  0.824939 -0.381920  0.544986  0.499586 -0.353760  0.392382 -0.236824   
120  0.715665 -0.550223  0.826875  0.799726 -0.485697  0.547577  0.031591   
121  0.594795 -0.495377  0.022718  0.013241 -0.445189  0.028243 -0.112142   
122  0.242819  0.062883  0.665258  0.736190  0.007943  0.431251  0.120598   
123 -0.753284 -0.546978  0.153732  0.228316 -0.483024  0.107128  0.350547   
124 -7.622581 -0.568484 -0.190614 -0.327078 -0.499920 -0.189675  0.228591   
125  0.790286 -0.568484 -0.051221  0.064444 -0.499920 -0.045962 -0.299402   
126  0.641693 -0.568484 -0.121817 -0.032053 -0.499920 -0.080096 -0.069105   
127  0.855329 -0.568484  0.256316  0.373059 -0.499920  0.145416  0.115238   
128  0.873747 -0.568484  0.111846  0.232071 -0.499920  0.029533 -0.023697   
129  0.404409 -0.568484  0.909496  0.921791 -0.499920  0.521449  0.231754   
130  0.104040  0.234788  0.838222  0.763350  0.137666  0.488096  0.148767   

           28    ...         4311      4312      4313      4314      4315  \
0   -0.221699    ...    -1.300000 -1.036887 -0.991674 -1.021653 -0.991608   
1   -0.133704    ...    -1.300000 -1.036887 -0.991674 -1.021653 -0.991608   
2   -0.101545    ...    -1.300000 -1.036887 -0.991674 -1.021653 -0.991608   
3   -0.085681    ...    -1.300000 -1.036887 -0.991674 -1.021653 -0.991608   
4   -0.107713    ...    -1.300000 -1.036887 -0.991674 -1.021653 -0.991608   
5   -0.100245    ...    -1.300000 -1.036887 -0.991674 -1.021653 -0.991608   
6   -0.161207    ...    -1.300000 -1.036887 -0.991674 -1.021653 -0.991608   
7   -0.193876    ...    -1.300000 -1.036887 -0.991674 -1.021653 -0.991608   
8   -0.158531    ...    -1.300000 -1.036887 -0.991674 -1.021653 -0.991608   
9   -0.166934    ...    -1.300000 -1.036887 -0.991674 -1.021653 -0.991608   
10  -0.170210    ...    -1.300000 -1.036887 -0.991674 -1.021653 -0.991608   
11  -0.186742    ...    -1.300000 -1.036887 -0.991674 -1.021653 -0.991608   
12  -0.199546    ...    -1.300000 -1.036887 -0.991674 -1.021653 -0.991608   
13  -0.169160    ...    -1.300000 -1.036887 -0.991674 -1.021653 -0.991608   
14  -0.174289    ...    -1.300000 -1.036887 -0.991674 -1.021653 -0.991608   
15  -0.104489    ...    -1.300000 -1.036887 -0.991674 -1.021653 -0.991608   
16  -0.205336    ...    -1.300000 -1.036887 -0.991674 -1.021653 -0.991608   
17  -0.189269    ...    -1.300000 -1.036887 -0.991674 -1.021653 -0.991608   
18  -0.156966    ...    -1.300000 -1.036887 -0.991674 -1.021653 -0.991608   
19  -0.197774    ...    -1.300000 -1.036887 -0.991674 -1.021653 -0.991608   
20  -0.176967    ...    -1.300000 -1.036887 -0.991674 -1.021653 -0.991608   
21  -0.175426    ...    -1.300000 -1.036887 -0.991674 -1.021653 -0.991608   
22  -0.255914    ...    -1.300000 -1.036887 -0.991674 -1.021653 -0.991608   
23  -0.099536    ...    -1.300000 -1.036887 -0.991674 -1.021653 -0.991608   
24  -0.186920    ...    -1.300000 -1.036887 -0.991674 -1.021653 -0.991608   
25  -0.201473    ...    -1.300000 -1.036887 -0.991674 -1.021653 -0.991608   
26  -0.290445    ...    -1.300000 -1.036887 -0.991674 -1.021653 -0.991608   
27   1.589547    ...    -1.300000 -1.036887 -0.991674 -1.021653 -0.991608   
28  -0.186074    ...    -1.300000 -1.036887 -0.991674 -1.021653 -0.991608   
29  -0.244937    ...    -1.300000 -1.036887 -0.991674 -1.021653 -0.991608   
..        ...    ...          ...       ...       ...       ...       ...   
101 -0.094708    ...     0.761415  0.895315  0.979186  0.617179  1.086850   
102 -0.187991    ...     0.805642  0.897419  0.967551  0.708609  1.098048   
103 -0.183907    ...     0.852889  0.897097  0.951590  0.821816  1.102186   
104 -0.183715    ...     0.895266  0.904293  0.945521  0.922058  1.099238   
105 -0.196927    ...     0.912609  0.916281  0.948135  1.039547  1.093995   
106 -0.178451    ...     0.846672  0.915111  0.942389  1.164177  1.095889   
107 -0.184081    ...     0.786231  0.909985  0.937919  1.270715  1.095723   
108 -0.194202    ...     0.736868  0.907632  0.936147  1.297961  1.092549   
109 -0.210954    ...     0.658923  0.910573  0.937739  1.316671  1.087452   
110 -0.220272    ...     0.588668  0.914929  0.942462  1.313575  1.083498   
111  5.202379    ...     0.531122  0.922988  0.951931  1.306676  1.080868   
112 -0.191934    ...     0.475720  0.930291  0.956570  1.286380  1.077502   
113 -0.272724    ...     0.479185  0.938027  0.963563  1.269510  1.071459   
114 -0.185526    ...     0.483264  0.947894  0.971050  1.238240  1.058017   
115 -0.117440    ...     0.473352  0.954076  0.976147  1.222465  1.048707   
116 -0.117279    ...     0.467208  0.958733  0.986059  1.208381  1.037299   
117 -0.170590    ...     0.471592  0.966222  0.990921  1.195225  1.030338   
118 -0.160196    ...     0.494199  0.966729  0.986089  1.202652  1.020146   
119 -0.128534    ...     0.506660  0.975086  0.997385  1.204428  1.011015   
120 -0.112119    ...     0.519224  0.984997  1.010336  1.201675  0.997274   
121 -0.181742    ...     0.547059  0.990947  1.021911  1.196272  0.989294   
122 -0.122126    ...     0.577041  0.995671  1.033017  1.173190  0.980361   
123 -0.158770    ...     0.590258  1.000748  1.035648  1.129194  0.973296   
124 -0.188754    ...     0.584838  1.009922  1.032677  1.100823  0.955869   
125 -0.208053    ...     0.581767  1.036627  1.034719  1.085007  0.948304   
126 -0.185659    ...     0.578357  1.063098  1.031508  1.059652  0.944139   
127 -0.182614    ...     0.575464  1.097791  1.027777  1.031796  0.942653   
128 -0.205313    ...     0.573261  1.132360  1.019764  0.993157  0.945072   
129 -0.102906    ...     0.572922  1.168444  1.027290  0.935189  0.942075   
130 -0.110286    ...    -1.300000  1.200532 -0.991674  0.914338 -0.991608   

         4316      4317      4318      4319      4320      4321      4322  \
0   -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994   
1   -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994   
2   -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994   
3   -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994   
4   -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994   
5   -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994   
6   -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994   
7   -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994   
8   -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994   
9   -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994   
10  -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994   
11  -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994   
12  -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994   
13  -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994   
14  -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994   
15  -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994   
16  -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994   
17  -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994   
18  -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994   
19  -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994   
20  -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994   
21  -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994   
22  -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994   
23  -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994   
24  -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994   
25  -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994   
26  -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994   
27  -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994   
28  -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994   
29  -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994   
..        ...       ...       ...       ...       ...       ...       ...   
101  0.987741  0.514887  0.975426  0.852340  0.751311  0.512499  0.152440   
102  0.891527  0.588986  0.971639  0.989928  0.777871  0.511422  0.240485   
103  0.787822  0.694703  0.971424  1.119715  0.813156  0.501295  0.429349   
104  0.727616  0.788585  0.961718  1.236148  0.844085  0.481413  0.642731   
105  0.713299  0.894299  0.956574  1.308199  0.905665  0.484429  0.837758   
106  0.675068  1.003277  0.950862  1.343988  0.952014  0.518475  0.980703   
107  0.644014  1.100489  0.948911  1.374585  1.000674  0.552345  1.130470   
108  0.643782  1.135972  0.950950  1.375228  1.051770  0.625149  1.262553   
109  0.703062  1.165722  0.952307  1.373026  1.073106  0.687372  1.343983   
110  0.733406  1.187909  0.960005  1.372714  1.107556  0.759476  1.417899   
111  0.761485  1.244170  0.966768  1.372824  1.163411  0.846829  1.481525   
112  0.782324  1.282711  0.970487  1.372767  1.185369  0.896952  1.505954   
113  0.827781  1.319238  0.976983  1.372819  1.204925  0.923020  1.527287   
114  0.884692  1.323212  0.982194  1.373086  1.199181  0.934144  1.499745   
115  0.971599  1.322388  0.992861  1.373033  1.182248  0.913554  1.460750   
116  1.079770  1.316875  1.001922  1.357765  1.164776  0.896952  1.408652   
117  1.161158  1.297923  1.004830  1.357487  1.165587  0.889877  1.399432   
118  1.218265  1.261877  1.005786  1.357701  1.164391  0.890257  1.393241   
119  1.343042  1.229553  1.005690  1.333742  1.146382  0.874135  1.337122   
120  1.471266  1.199781  1.003107  1.246810  1.124848  0.857498  1.271884   
121  1.558683  1.187333  1.009092  1.085649  1.139357  0.881206  1.261494   
122  1.662889  1.170188  1.004441  0.861397  1.147749  0.911076  1.233655   
123  1.743964  1.141376  1.000174  0.655738  1.136348  0.907971  1.183771   
124  1.867913  1.130348  0.995892  0.410991  1.100740  0.883591  1.069320   
125  1.900589  1.127729  0.998131  0.103462  1.117973  0.877266  1.057818   
126  1.951052  1.118981  1.002292 -0.198036  1.133604  0.882805  1.039624   
127  2.057439  1.108187  1.006347 -0.436724  1.146472  0.875983  1.011781   
128  2.053308  1.075614  1.005827 -0.601559  1.152569  0.851412  1.011956   
129  2.091451  1.021860  1.008149 -0.703088  1.145196  0.833326  0.966913   
130 -0.897575  1.003463 -0.992045 -0.903636  1.148980 -1.299491  0.957097   

         4323      4324      4325      4326      4327      4328      4329  \
0   -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988   
1   -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988   
2   -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988   
3   -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988   
4   -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988   
5   -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988   
6   -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988   
7   -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988   
8   -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988   
9   -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988   
10  -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988   
11  -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988   
12  -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988   
13  -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988   
14  -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988   
15  -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988   
16  -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988   
17  -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988   
18  -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988   
19  -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988   
20  -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988   
21  -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988   
22  -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988   
23  -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988   
24  -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988   
25  -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988   
26  -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988   
27  -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988   
28  -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988   
29  -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988   
..        ...       ...       ...       ...       ...       ...       ...   
101  0.924107  1.299071  1.231534  1.020582  0.747029  0.722886  0.809215   
102  0.965998  1.307196  1.203327  1.011123  0.449254  0.649353  0.766987   
103  1.025335  1.350142  1.152890  0.989969  0.786030  0.632033  0.922312   
104  1.064819  1.382306  1.139304  0.985695  1.289130  0.727184  1.023473   
105  1.077952  1.367314  1.143855  0.985765  1.122053  0.893260  1.118280   
106  0.999576  1.405106  1.119387  0.978534  1.103160  1.149989  1.245052   
107  0.933066  1.402812  1.118289  0.976016  1.187710  1.373575  1.397867   
108  0.874036  1.383816  1.098748  0.971498  1.461453  1.635945  1.565446   
109  0.792693  1.390360  1.084691  0.972771  1.903505  1.890322  1.738836   
110  0.723736  1.371702  1.088652  0.975252  1.879084  1.949017  1.731190   
111  0.665836  1.355264  1.070740  0.976517  1.546954  1.855479  1.764962   
112  0.607421  1.393601  1.034318  0.978632  2.115828  1.687225  1.759499   
113  0.603043  1.382253  0.978676  0.976987  2.226667  1.368102  1.763305   
114  0.583536  1.360821  0.963281  0.981484  2.375056  1.007451  1.730987   
115  0.571128  1.358303  0.952569  0.989179  2.047167  0.615043  1.764461   
116  0.545632  1.324991  0.951146  0.999509  1.685932  0.399944  1.599606   
117  0.550988  1.316453  0.900924  1.001908  2.077410  0.359158  1.461671   
118  0.576477  1.297063  0.846523  0.996625  2.651805  0.355279  1.331906   
119  0.580475  1.282397  0.838986  1.008356  2.013802  0.357856  1.255270   
120  0.585151  1.250100  0.897888  1.031206  1.495501  0.500016  1.162414   
121  0.626995  1.236444  0.843934  1.031653  1.204577  0.713956  1.086009   
122  0.638088  1.172866  0.874622  1.037555  1.140265  0.928292  0.977448   
123  0.623304  1.126065  0.955397  1.042265  1.061346  0.981875  1.029609   
124  0.520692  1.061314  1.141393  1.059022  0.587315  1.077188  1.121039   
125  0.468945  1.060710  1.263109  1.080595  0.215169  1.104631  1.131481   
126  0.396489  1.017658  1.396629  1.095748  0.221711  1.133707  1.134086   
127  0.304313  0.972793  1.571572  1.110736  0.194014  1.139673  1.164964   
128  0.235881  0.994969  1.763314  1.132965  0.220057  1.251421  1.213085   
129  0.158429  1.043038  1.871936  1.153859  0.438814  1.336613  1.231104   
130 -1.285899  1.087215 -0.965811 -0.991597  0.697918  1.554784 -0.929988   

         4330      4331      4332      4333      4334      4335      4336  \
0   -0.980759 -0.730907 -0.991360 -0.774115 -0.724674 -0.542333 -0.992039   
1   -0.980759 -0.730907 -0.991360 -0.774115 -0.724674 -0.542333 -0.992039   
2   -0.980759 -0.730907 -0.991360 -0.774115 -0.724674 -0.542333 -0.992039   
3   -0.980759 -0.730907 -0.991360 -0.774115 -0.724674 -0.542333 -0.992039   
4   -0.980759 -0.730907 -0.991360 -0.774115 -0.724674 -0.542333 -0.992039   
5   -0.980759 -0.730907 -0.991360 -0.774115 -0.724674 -0.542333 -0.992039   
6   -0.980759 -0.730907 -0.991360 -0.774115 -0.724674 -0.542333 -0.992039   
7   -0.980759 -0.730907 -0.991360 -0.774115 -0.724674 -0.542333 -0.992039   
8   -0.980759 -0.730907 -0.991360 -0.774115 -0.724674 -0.542333 -0.992039   
9   -0.980759 -0.730907 -0.991360 -0.774115 -0.724674 -0.542333 -0.992039   
10  -0.980759 -0.730907 -0.991360 -0.774115 -0.724674 -0.542333 -0.992039   
11  -0.980759 -0.730907 -0.991360 -0.774115 -0.724674 -0.542333 -0.992039   
12  -0.980759 -0.730907 -0.991360 -0.774115 -0.724674 -0.542333 -0.992039   
13  -0.980759 -0.730907 -0.991360 -0.774115 -0.724674 -0.542333 -0.992039   
14  -0.980759 -0.730907 -0.991360 -0.774115 -0.724674 -0.542333 -0.992039   
15  -0.980759 -0.730907 -0.991360 -0.774115 -0.724674 -0.542333 -0.992039   
16  -0.980759 -0.730907 -0.991360 -0.774115 -0.724674 -0.542333 -0.992039   
17  -0.980759 -0.730907 -0.991360 -0.774115 -0.724674 -0.542333 -0.992039   
18  -0.980759 -0.730907 -0.991360 -0.774115 -0.724674 -0.542333 -0.992039   
19  -0.980759 -0.730907 -0.991360 -0.774115 -0.724674 -0.542333 -0.992039   
20  -0.980759 -0.730907 -0.991360 -0.774115 -0.724674 -0.542333 -0.992039   
21  -0.980759 -0.730907 -0.991360 -0.774115 -0.724674 -0.542333 -0.992039   
22  -0.980759 -0.730907 -0.991360 -0.774115 -0.724674 -0.542333 -0.992039   
23  -0.980759 -0.730907 -0.991360 -0.774115 -0.724674 -0.542333 -0.992039   
24  -0.980759 -0.730907 -0.991360 -0.774115 -0.724674 -0.542333 -0.992039   
25  -0.980759 -0.730907 -0.991360 -0.774115 -0.724674 -0.542333 -0.992039   
26  -0.980759 -0.730907 -0.991360 -0.774115 -0.724674 -0.542333 -0.992039   
27  -0.980759 -0.730907 -0.991360 -0.774115 -0.724674 -0.542333 -0.992039   
28  -0.980759 -0.730907 -0.991360 -0.774115 -0.724674 -0.542333 -0.992039   
29  -0.980759 -0.730907 -0.991360 -0.774115 -0.724674 -0.542333 -0.992039   
..        ...       ...       ...       ...       ...       ...       ...   
101  0.649813  1.764333  0.988939  1.743413  1.437082  2.594316  0.974960   
102  0.671871  1.823376  0.977826  1.750169  1.241295  2.487819  0.976249   
103  0.785261  1.891404  0.954900  1.038140  0.749037  0.790711  0.979060   
104  0.913266  1.936779  0.944705  0.273372  0.312800  0.089672  0.986086   
105  1.051904  1.900214  0.940239  0.019011  0.091372 -0.113075  0.991401   
106  1.196582  1.827844  0.936402 -0.002932  0.102505 -0.131627  0.988013   
107  1.293182  1.744187  0.932820 -0.022204  0.046807 -0.119241  0.981956   
108  1.366656  1.565344  0.927847 -0.024344 -0.024341 -0.089888  0.979248   
109  1.386134  1.407662  0.928311 -0.008756  0.004943 -0.059463  0.981007   
110  1.306095  1.341985  0.928134 -0.026106 -0.034496 -0.061500  0.985427   
111  1.241730  1.320862  0.928657 -0.106163 -0.160445 -0.107900  0.990999   
112  1.169435  1.348609  0.933523 -0.133414 -0.178995 -0.128964  0.996821   
113  1.111046  1.373928  0.937796 -0.168114 -0.202833 -0.161069  1.002636   
114  1.076998  1.432876  0.945301 -0.178247 -0.175307 -0.216664  1.008869   
115  1.096334  1.547275  0.956765 -0.159464 -0.083716 -0.231205  1.010812   
116  1.103419  1.761430  0.972135 -0.087858  0.049136 -0.192791  1.005533   
117  1.130761  1.758613  0.980749 -0.092750  0.067560 -0.168373  1.002226   
118  1.114704  1.651174  0.980793 -0.077902  0.106054 -0.134797  1.000554   
119  1.111690  1.644885  0.997163  0.021235  0.231908 -0.037814  1.000165   
120  1.055678  1.683457  1.014034  0.150554  0.355701  0.055856  0.997361   
121  0.996406  1.499999  1.018873  0.164671  0.345006  0.074642  0.991527   
122  0.907764  1.311926  1.021408  0.232941  0.362067  0.102493  0.984096   
123  0.926970  1.191082  1.025989  0.540704  0.772031  0.229773  0.985113   
124  0.938021  1.214402  1.031546  0.988749  1.426077  0.388349  0.995412   
125  1.011140  1.086619  1.031050  1.040065  1.524318  0.394928  1.025282   
126  1.047492  0.958984  1.027187  1.043735  1.607720  0.351643  1.057848   
127  1.116951  0.791446  1.028727  1.014524  1.824807  0.318781  1.099188   
128  1.217198  0.735115  1.029184  1.044299  0.732369  0.337120  1.143518   
129  1.287958  0.645227  1.041989  1.150565  0.815745  0.384612  1.187765   
130 -0.980759 -0.730907 -0.991360  1.068414 -0.724674  0.386679 -0.992039   

         4337      4338      4339      4340  
0   -1.036513 -0.981902 -0.995691 -0.800299  
1   -1.036513 -0.981902 -0.995691 -0.800299  
2   -1.036513 -0.981902 -0.995691 -0.800299  
3   -1.036513 -0.981902 -0.995691 -0.800299  
4   -1.036513 -0.981902 -0.995691 -0.800299  
5   -1.036513 -0.981902 -0.995691 -0.800299  
6   -1.036513 -0.981902 -0.995691 -0.800299  
7   -1.036513 -0.981902 -0.995691 -0.800299  
8   -1.036513 -0.981902 -0.995691 -0.800299  
9   -1.036513 -0.981902 -0.995691 -0.800299  
10  -1.036513 -0.981902 -0.995691 -0.800299  
11  -1.036513 -0.981902 -0.995691 -0.800299  
12  -1.036513 -0.981902 -0.995691 -0.800299  
13  -1.036513 -0.981902 -0.995691 -0.800299  
14  -1.036513 -0.981902 -0.995691 -0.800299  
15  -1.036513 -0.981902 -0.995691 -0.800299  
16  -1.036513 -0.981902 -0.995691 -0.800299  
17  -1.036513 -0.981902 -0.995691 -0.800299  
18  -1.036513 -0.981902 -0.995691 -0.800299  
19  -1.036513 -0.981902 -0.995691 -0.800299  
20  -1.036513 -0.981902 -0.995691 -0.800299  
21  -1.036513 -0.981902 -0.995691 -0.800299  
22  -1.036513 -0.981902 -0.995691 -0.800299  
23  -1.036513 -0.981902 -0.995691 -0.800299  
24  -1.036513 -0.981902 -0.995691 -0.800299  
25  -1.036513 -0.981902 -0.995691 -0.800299  
26  -1.036513 -0.981902 -0.995691 -0.800299  
27  -1.036513 -0.981902 -0.995691 -0.800299  
28  -1.036513 -0.981902 -0.995691 -0.800299  
29  -1.036513 -0.981902 -0.995691 -0.800299  
..        ...       ...       ...       ...  
101  0.840711  0.744666  0.643472  1.205978  
102  0.837712  0.742849  0.578941  0.932542  
103  0.839520  0.743441  0.517883  0.920269  
104  0.847406  0.753197  0.483486  1.041444  
105  0.857533  0.768184  0.517707  0.941386  
106  0.864455  0.774745  0.552730  1.049873  
107  0.868167  0.776790  0.603959  1.165573  
108  0.865820  0.771937  0.652789  1.279808  
109  0.866945  0.770830  0.697130  1.557481  
110  0.872018  0.774221  0.743726  1.709151  
111  0.877023  0.780697  0.792123  1.583769  
112  0.883113  0.786832  0.819010  1.560670  
113  0.887605  0.790673  0.848832  1.559894  
114  0.886210  0.791394  0.895737  1.564819  
115  0.886925  0.797362  0.962674  1.526159  
116  0.895019  0.807328  1.056537  1.412854  
117  0.904051  0.821839  1.132846  1.421087  
118  0.914263  0.835388  1.188078  1.589667  
119  0.929064  0.851919  1.276414  1.695728  
120  0.944779  0.870493  1.385299  1.713457  
121  0.970153  0.892549  1.416809  1.529177  
122  0.991997  0.908414  1.484227  1.309775  
123  1.009875  0.921618  1.537728  1.151473  
124  1.029284  0.934298  1.615405  1.004163  
125  1.045781  0.944189  1.600086  0.843282  
126  1.058542  0.952315  1.633981  0.767760  
127  1.071676  0.959615  1.678787  0.507489  
128  1.086924  0.962711  1.638107  0.694722  
129  1.100510  0.970256  1.562026  0.929974  
130  1.114567 -0.981902  1.401969  1.133826  

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C:\Users\Daniela\Anaconda3.5\lib\site-packages\numpy\lib\function_base.py:3652: RuntimeWarning: Invalid value encountered in percentile
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Entering maxim
Entering maxim
Entering maxim
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Entering maxim
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colnames  2883
Entering maxim
Entering maxim
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Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
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Entering maxim
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Entering maxim
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Entering maxim
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Entering maxim
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colnames  2923
Entering maxim
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Entering maxim
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Entering maxim
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Entering maxim
Entering maxim
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Entering maxim
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Entering maxim
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Entering maxim
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Entering maxim
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Entering maxim
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Entering maxim
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Entering maxim
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Entering maxim
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Entering maxim
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Entering maxim
enterinf minimum
Entering maxim
Entering maxim
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3116
enterinf minimum
enterinf minimum
colnames  3117
colnames  3118
colnames  3119
colnames  3120
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3121
colnames  3122
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3123
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3124
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3125
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3126
enterinf minimum
enterinf minimum
colnames  3127
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3128
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3129
colnames  3130
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3131
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3132
Entering maxim
Entering maxim
Entering maxim
Entering maxim
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Entering maxim
Entering maxim
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Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3133
colnames  3134
colnames  3135
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3136
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3137
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
enterinf minimum
Entering maxim
enterinf minimum
Entering maxim
colnames  3138
colnames  3139
colnames  3140
Entering maxim
Entering maxim
colnames  3141
colnames  3142
colnames  3143
Entering maxim
Entering maxim
colnames  3144
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3145
Entering maxim
Entering maxim
Entering maxim
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Entering maxim
Entering maxim
Entering maxim
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Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3146
colnames  3147
colnames  3148
colnames  3149
Entering maxim
colnames  3150
Entering maxim
Entering maxim
colnames  3151
colnames  3152
colnames  3153
colnames  3154
colnames  3155
Entering maxim
Entering maxim
Entering maxim
colnames  3156
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3157
colnames  3158
colnames  3159
Entering maxim
Entering maxim
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3160
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enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3161
colnames  3162
enterinf minimum
colnames  3163
colnames  3164
Entering maxim
colnames  3165
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3166
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3167
colnames  3168
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3169
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3170
colnames  3171
colnames  3172
colnames  3173
colnames  3174
colnames  3175
colnames  3176
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
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Entering maxim
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Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3177
enterinf minimum
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enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3178
colnames  3179
colnames  3180
Entering maxim
colnames  3181
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3182
colnames  3183
colnames  3184
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3185
enterinf minimum
colnames  3186
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3187
enterinf minimum
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colnames  3188
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3189
Entering maxim
Entering maxim
Entering maxim
colnames  3190
Entering maxim
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Entering maxim
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Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
colnames  3191
enterinf minimum
enterinf minimum
enterinf minimum
Entering maxim
colnames  3192
colnames  3193
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3194
enterinf minimum
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colnames  3195
colnames  3196
colnames  3197
colnames  3198
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3199
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
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colnames  3200
colnames  3201
colnames  3202
colnames  3203
enterinf minimum
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enterinf minimum
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colnames  3204
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3205
colnames  3206
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3207
colnames  3208
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colnames  3210
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colnames  3211
enterinf minimum
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3212
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colnames  3214
colnames  3215
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colnames  3217
Entering maxim
Entering maxim
Entering maxim
colnames  3218
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enterinf minimum
enterinf minimum
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enterinf minimum
enterinf minimum
enterinf minimum
colnames  3219
colnames  3220
colnames  3221
colnames  3222
colnames  3223
colnames  3224
enterinf minimum
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enterinf minimum
enterinf minimum
enterinf minimum
colnames  3225
colnames  3226
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3227
colnames  3228
colnames  3229
colnames  3230
colnames  3231
colnames  3232
Entering maxim
colnames  3233
Entering maxim
colnames  3234
Entering maxim
Entering maxim
Entering maxim
colnames  3235
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3236
Entering maxim
Entering maxim
Entering maxim
colnames  3237
colnames  3238
Entering maxim
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Entering maxim
colnames  3239
Entering maxim
Entering maxim
enterinf minimum
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enterinf minimum
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colnames  3240
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3241
colnames  3242
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enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
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colnames  3243
colnames  3244
colnames  3245
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colnames  3246
enterinf minimum
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colnames  3247
enterinf minimum
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colnames  3248
enterinf minimum
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colnames  3249
enterinf minimum
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colnames  3250
colnames  3251
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colnames  3252
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colnames  3253
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3254
enterinf minimum
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Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
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colnames  3255
enterinf minimum
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Entering maxim
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colnames  3256
colnames  3257
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colnames  3258
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colnames  3259
Entering maxim
Entering maxim
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Entering maxim
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colnames  3260
Entering maxim
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Entering maxim
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Entering maxim
enterinf minimum
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colnames  3261
Entering maxim
Entering maxim
Entering maxim
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colnames  3262
colnames  3263
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
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Entering maxim
Entering maxim
Entering maxim
enterinf minimum
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colnames  3264
enterinf minimum
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colnames  3265
enterinf minimum
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colnames  3266
enterinf minimum
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colnames  3267
Entering maxim
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colnames  3268
enterinf minimum
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colnames  3269
enterinf minimum
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colnames  3270
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colnames  3271
colnames  3272
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colnames  3273
Entering maxim
Entering maxim
Entering maxim
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Entering maxim
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Entering maxim
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colnames  3275
Entering maxim
Entering maxim
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colnames  3276
Entering maxim
Entering maxim
Entering maxim
Entering maxim
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Entering maxim
Entering maxim
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Entering maxim
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Entering maxim
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colnames  3278
Entering maxim
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colnames  3282
colnames  3283
Entering maxim
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colnames  3284
Entering maxim
Entering maxim
Entering maxim
Entering maxim
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colnames  3285
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Entering maxim
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colnames  3287
Entering maxim
Entering maxim
Entering maxim
colnames  3288
Entering maxim
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Entering maxim
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Entering maxim
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Entering maxim
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colnames  3299
Entering maxim
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Entering maxim
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Entering maxim
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Entering maxim
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Entering maxim
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Entering maxim
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colnames  3304
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Entering maxim
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Entering maxim
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Entering maxim
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Entering maxim
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Entering maxim
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Entering maxim
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Entering maxim
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Entering maxim
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Entering maxim
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Entering maxim
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Entering maxim
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enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3581
colnames  3582
colnames  3583
colnames  3584
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
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enterinf minimum
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colnames  3585
Entering maxim
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Entering maxim
enterinf minimum
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colnames  3586
Entering maxim
Entering maxim
Entering maxim
Entering maxim
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Entering maxim
Entering maxim
enterinf minimum
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colnames  3587
Entering maxim
Entering maxim
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colnames  3588
Entering maxim
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Entering maxim
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Entering maxim
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Entering maxim
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colnames  3589
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colnames  3590
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colnames  3591
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colnames  3592
colnames  3593
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colnames  3594
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
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colnames  3595
colnames  3596
enterinf minimum
Entering maxim
enterinf minimum
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colnames  3597
Entering maxim
Entering maxim
colnames  3598
Entering maxim
Entering maxim
enterinf minimum
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colnames  3599
colnames  3600
colnames  3601
colnames  3602
Entering maxim
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colnames  3603
Entering maxim
Entering maxim
colnames  3604
enterinf minimum
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colnames  3605
colnames  3606
Entering maxim
Entering maxim
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Entering maxim
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colnames  3608
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colnames  3609
Entering maxim
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Entering maxim
Entering maxim
colnames  3610
colnames  3611
Entering maxim
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Entering maxim
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Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
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colnames  3612
Entering maxim
Entering maxim
Entering maxim
Entering maxim
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colnames  3613
Entering maxim
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Entering maxim
enterinf minimum
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Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
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colnames  3614
enterinf minimum
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colnames  3615
enterinf minimum
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colnames  3616
enterinf minimum
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colnames  3617
colnames  3618
colnames  3619
Entering maxim
enterinf minimum
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Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
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Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
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colnames  3620
Entering maxim
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Entering maxim
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colnames  3621
colnames  3622
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Entering maxim
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Entering maxim
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Entering maxim
enterinf minimum
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enterinf minimum
enterinf minimum
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colnames  3625
colnames  3626
colnames  3627
Entering maxim
Entering maxim
Entering maxim
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colnames  3628
Entering maxim
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colnames  3629
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colnames  3630
Entering maxim
Entering maxim
colnames  3631
enterinf minimum
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Entering maxim
enterinf minimum
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colnames  3632
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colnames  3633
Entering maxim
Entering maxim
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colnames  3634
enterinf minimum
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colnames  3635
Entering maxim
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Entering maxim
Entering maxim
Entering maxim
enterinf minimum
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colnames  3636
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
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Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
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Entering maxim
enterinf minimum
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colnames  3637
enterinf minimum
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colnames  3638
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colnames  3639
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colnames  3640
Entering maxim
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colnames  3641
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colnames  3642
Entering maxim
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colnames  3643
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colnames  3644
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colnames  3645
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colnames  3646
colnames  3647
Entering maxim
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Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
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colnames  3648
enterinf minimum
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colnames  3649
enterinf minimum
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colnames  3650
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colnames  3651
Entering maxim
Entering maxim
Entering maxim
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colnames  3652
enterinf minimum
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colnames  3653
enterinf minimum
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colnames  3654
Entering maxim
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Entering maxim
Entering maxim
Entering maxim
enterinf minimum
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colnames  3655
enterinf minimum
Entering maxim
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colnames  3656
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colnames  3657
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colnames  3659
enterinf minimum
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colnames  3660
Entering maxim
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colnames  3661
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colnames  3663
colnames  3664
Entering maxim
Entering maxim
Entering maxim
Entering maxim
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colnames  3665
colnames  3666
enterinf minimum
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Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
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Entering maxim
enterinf minimum
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colnames  3670
colnames  3671
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colnames  3673
colnames  3674
colnames  3675
colnames  3676
colnames  3677
Entering maxim
Entering maxim
colnames  3678
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colnames  3681
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Entering maxim
Entering maxim
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colnames  3685
Entering maxim
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colnames  3686
Entering maxim
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colnames  3687
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Entering maxim
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colnames  3714
Entering maxim
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colnames  3715
Entering maxim
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colnames  3716
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Entering maxim
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colnames  3723
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colnames  3728
Entering maxim
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colnames  3733
Entering maxim
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Entering maxim
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colnames  3734
colnames  3735
Entering maxim
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enterinf minimum
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
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Entering maxim
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Entering maxim
colnames  3736
enterinf minimum
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colnames  3737
colnames  3738
colnames  3739
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Entering maxim
Entering maxim
Entering maxim
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Entering maxim
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colnames  3741
colnames  3742
Entering maxim
Entering maxim
Entering maxim
Entering maxim
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Entering maxim
colnames  3743
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enterinf minimum
colnames  3745
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3746
colnames  3747
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3748
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3749
colnames  3750
enterinf minimum
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Entering maxim
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colnames  3752
Entering maxim
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colnames  3753
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enterinf minimum
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Entering maxim
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Entering maxim
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Entering maxim
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Entering maxim
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Entering maxim
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colnames  3769
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Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
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enterinf minimum
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Entering maxim
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Entering maxim
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Entering maxim
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colnames  3781
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enterinf minimum
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Entering maxim
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colnames  3783
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Entering maxim
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Entering maxim
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Entering maxim
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colnames  3790
colnames  3791
enterinf minimum
colnames  3792
colnames  3793
colnames  3794
colnames  3795
colnames  3796
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3797
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3798
colnames  3799
colnames  3800
Entering maxim
colnames  3801
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3802
colnames  3803
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3804
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3805
colnames  3806
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3807
enterinf minimum
Entering maxim
colnames  3808
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3809
Entering maxim
colnames  3810
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3811
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
Entering maxim
colnames  3812
colnames  3813
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3814
enterinf minimum
colnames  3815
colnames  3816
colnames  3817
Entering maxim
colnames  3818
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3819
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
Entering maxim
Entering maxim
colnames  3820
colnames  3821
colnames  3822
colnames  3823
enterinf minimum
enterinf minimum
colnames  3824
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3825
colnames  3826
Entering maxim
colnames  3827
colnames  3828
colnames  3829
colnames  3830
enterinf minimum
enterinf minimum
colnames  3831
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3832
colnames  3833
colnames  3834
colnames  3835
colnames  3836
colnames  3837
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3838
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3839
colnames  3840
colnames  3841
colnames  3842
colnames  3843
Entering maxim
colnames  3844
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3845
colnames  3846
colnames  3847
colnames  3848
colnames  3849
colnames  3850
colnames  3851
colnames  3852
colnames  3853
colnames  3854
Entering maxim
Entering maxim
Entering maxim
colnames  3855
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3856
colnames  3857
colnames  3858
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3859
enterinf minimum
enterinf minimum
Entering maxim
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3860
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3861
colnames  3862
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3863
colnames  3864
colnames  3865
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3866
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3867
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3868
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3869
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3870
colnames  3871
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3872
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3873
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3874
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3875
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3876
colnames  3877
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3878
enterinf minimum
enterinf minimum
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enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3879
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3880
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3881
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3882
colnames  3883
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3884
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
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enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
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colnames  3885
enterinf minimum
enterinf minimum
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enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
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colnames  3886
enterinf minimum
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enterinf minimum
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enterinf minimum
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enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
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colnames  3887
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
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enterinf minimum
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enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
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enterinf minimum
enterinf minimum
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colnames  3888
enterinf minimum
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enterinf minimum
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enterinf minimum
enterinf minimum
enterinf minimum
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enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
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colnames  3889
enterinf minimum
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enterinf minimum
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enterinf minimum
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enterinf minimum
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enterinf minimum
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enterinf minimum
enterinf minimum
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colnames  3890
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
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enterinf minimum
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enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
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enterinf minimum
enterinf minimum
enterinf minimum
colnames  3891
colnames  3892
enterinf minimum
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enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3893
Entering maxim
colnames  3894
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
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enterinf minimum
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enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
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enterinf minimum
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enterinf minimum
enterinf minimum
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colnames  3895
Entering maxim
enterinf minimum
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enterinf minimum
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enterinf minimum
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Entering maxim
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
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enterinf minimum
enterinf minimum
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colnames  3896
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3897
Entering maxim
Entering maxim
colnames  3898
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3899
enterinf minimum
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enterinf minimum
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enterinf minimum
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enterinf minimum
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enterinf minimum
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enterinf minimum
enterinf minimum
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colnames  3900
enterinf minimum
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colnames  3901
enterinf minimum
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enterinf minimum
enterinf minimum
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enterinf minimum
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colnames  3902
colnames  3903
colnames  3904
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3905
colnames  3906
colnames  3907
enterinf minimum
enterinf minimum
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
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enterinf minimum
enterinf minimum
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enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
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colnames  3908
Entering maxim
Entering maxim
enterinf minimum
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enterinf minimum
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enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
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enterinf minimum
enterinf minimum
enterinf minimum
colnames  3909
colnames  3910
colnames  3911
colnames  3912
Entering maxim
enterinf minimum
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enterinf minimum
enterinf minimum
enterinf minimum
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colnames  3913
Entering maxim
colnames  3914
enterinf minimum
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enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
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enterinf minimum
enterinf minimum
colnames  3915
colnames  3916
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3917
enterinf minimum
enterinf minimum
enterinf minimum
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enterinf minimum
enterinf minimum
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enterinf minimum
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enterinf minimum
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enterinf minimum
enterinf minimum
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colnames  3918
Entering maxim
enterinf minimum
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enterinf minimum
enterinf minimum
enterinf minimum
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enterinf minimum
enterinf minimum
enterinf minimum
colnames  3919
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3920
Entering maxim
colnames  3921
Entering maxim
enterinf minimum
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enterinf minimum
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Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
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enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
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colnames  3922
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3923
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
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Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
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colnames  3924
enterinf minimum
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colnames  3925
enterinf minimum
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colnames  3926
enterinf minimum
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colnames  3927
Entering maxim
Entering maxim
colnames  3928
colnames  3929
enterinf minimum
enterinf minimum
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enterinf minimum
enterinf minimum
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
Entering maxim
enterinf minimum
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Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3930
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3931
Entering maxim
colnames  3932
colnames  3933
colnames  3934
colnames  3935
colnames  3936
colnames  3937
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3938
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3939
enterinf minimum
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enterinf minimum
enterinf minimum
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enterinf minimum
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colnames  3940
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3941
enterinf minimum
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colnames  3942
enterinf minimum
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colnames  3943
Entering maxim
Entering maxim
enterinf minimum
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colnames  3944
enterinf minimum
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colnames  3945
Entering maxim
enterinf minimum
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Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
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colnames  3946
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
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Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
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colnames  3947
enterinf minimum
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colnames  3948
enterinf minimum
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colnames  3949
enterinf minimum
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colnames  3950
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
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Entering maxim
Entering maxim
Entering maxim
colnames  3951
enterinf minimum
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colnames  3952
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
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Entering maxim
Entering maxim
colnames  3953
enterinf minimum
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colnames  3954
enterinf minimum
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colnames  3955
enterinf minimum
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colnames  3956
colnames  3957
Entering maxim
Entering maxim
enterinf minimum
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Entering maxim
Entering maxim
Entering maxim
enterinf minimum
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colnames  3958
enterinf minimum
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enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3959
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  3960
enterinf minimum
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enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
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colnames  3961
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
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enterinf minimum
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enterinf minimum
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enterinf minimum
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colnames  3962
enterinf minimum
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colnames  3963
enterinf minimum
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enterinf minimum
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enterinf minimum
enterinf minimum
enterinf minimum
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enterinf minimum
enterinf minimum
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colnames  3964
Entering maxim
Entering maxim
enterinf minimum
enterinf minimum
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enterinf minimum
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
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enterinf minimum
enterinf minimum
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colnames  3965
Entering maxim
enterinf minimum
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colnames  3966
enterinf minimum
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colnames  3967
colnames  3968
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colnames  3969
enterinf minimum
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colnames  3970
Entering maxim
enterinf minimum
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enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
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colnames  3971
colnames  3972
colnames  3973
colnames  3974
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
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enterinf minimum
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enterinf minimum
enterinf minimum
enterinf minimum
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colnames  3975
colnames  3976
enterinf minimum
enterinf minimum
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enterinf minimum
enterinf minimum
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Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
colnames  3977
colnames  3978
colnames  3979
colnames  3980
colnames  3981
colnames  3982
colnames  3983
colnames  3984
colnames  3985
colnames  3986
colnames  3987
colnames  3988
colnames  3989
colnames  3990
colnames  3991
colnames  3992
colnames  3993
colnames  3994
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3995
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
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Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3996
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  3997
colnames  3998
colnames  3999
colnames  4000
colnames  4001
colnames  4002
colnames  4003
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colnames  4005
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colnames  4007
colnames  4008
colnames  4009
colnames  4010
colnames  4011
colnames  4012
colnames  4013
colnames  4014
colnames  4015
colnames  4016
colnames  4017
colnames  4018
colnames  4019
colnames  4020
colnames  4021
colnames  4022
colnames  4023
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4024
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
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Entering maxim
colnames  4025
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4026
colnames  4027
colnames  4028
colnames  4029
colnames  4030
colnames  4031
colnames  4032
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
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Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4033
colnames  4034
colnames  4035
colnames  4036
colnames  4037
colnames  4038
Entering maxim
Entering maxim
colnames  4039
enterinf minimum
colnames  4040
colnames  4041
colnames  4042
Entering maxim
enterinf minimum
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enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4043
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4044
colnames  4045
Entering maxim
Entering maxim
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
colnames  4046
enterinf minimum
colnames  4047
colnames  4048
colnames  4049
colnames  4050
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
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Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4051
colnames  4052
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4053
colnames  4054
colnames  4055
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4056
colnames  4057
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4058
enterinf minimum
enterinf minimum
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4059
colnames  4060
enterinf minimum
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
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Entering maxim
Entering maxim
Entering maxim
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Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4061
enterinf minimum
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enterinf minimum
enterinf minimum
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enterinf minimum
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enterinf minimum
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colnames  4062
Entering maxim
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colnames  4063
colnames  4064
enterinf minimum
Entering maxim
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colnames  4065
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4066
colnames  4067
Entering maxim
Entering maxim
Entering maxim
colnames  4068
colnames  4069
colnames  4070
Entering maxim
colnames  4071
colnames  4072
colnames  4073
colnames  4074
colnames  4075
Entering maxim
Entering maxim
Entering maxim
colnames  4076
colnames  4077
colnames  4078
enterinf minimum
enterinf minimum
Entering maxim
colnames  4079
colnames  4080
Entering maxim
Entering maxim
colnames  4081
colnames  4082
colnames  4083
colnames  4084
colnames  4085
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4086
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
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Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
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colnames  4087
colnames  4088
colnames  4089
Entering maxim
Entering maxim
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Entering maxim
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Entering maxim
Entering maxim
Entering maxim
colnames  4090
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enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
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enterinf minimum
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colnames  4091
colnames  4092
colnames  4093
colnames  4094
colnames  4095
Entering maxim
Entering maxim
Entering maxim
Entering maxim
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Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4096
enterinf minimum
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colnames  4097
colnames  4098
Entering maxim
Entering maxim
colnames  4099
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4100
colnames  4101
enterinf minimum
colnames  4102
colnames  4103
colnames  4104
colnames  4105
colnames  4106
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
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Entering maxim
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Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4107
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4108
colnames  4109
colnames  4110
Entering maxim
colnames  4111
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4112
colnames  4113
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4114
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4115
colnames  4116
Entering maxim
Entering maxim
Entering maxim
colnames  4117
enterinf minimum
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enterinf minimum
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colnames  4118
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
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colnames  4119
colnames  4120
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
enterinf minimum
colnames  4121
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4122
colnames  4123
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
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Entering maxim
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Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4124
colnames  4125
colnames  4126
colnames  4127
colnames  4128
enterinf minimum
enterinf minimum
enterinf minimum
Entering maxim
enterinf minimum
enterinf minimum
Entering maxim
colnames  4129
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
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Entering maxim
colnames  4130
colnames  4131
colnames  4132
colnames  4133
enterinf minimum
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colnames  4134
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4135
colnames  4136
colnames  4137
colnames  4138
colnames  4139
colnames  4140
enterinf minimum
colnames  4141
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4142
colnames  4143
Entering maxim
Entering maxim
colnames  4144
colnames  4145
colnames  4146
colnames  4147
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4148
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4149
colnames  4150
colnames  4151
colnames  4152
colnames  4153
Entering maxim
Entering maxim
Entering maxim
colnames  4154
enterinf minimum
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enterinf minimum
colnames  4155
colnames  4156
Entering maxim
colnames  4157
colnames  4158
colnames  4159
colnames  4160
colnames  4161
colnames  4162
colnames  4163
colnames  4164
Entering maxim
colnames  4165
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4166
colnames  4167
colnames  4168
Entering maxim
Entering maxim
Entering maxim
colnames  4169
enterinf minimum
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Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
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colnames  4170
Entering maxim
Entering maxim
Entering maxim
Entering maxim
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Entering maxim
Entering maxim
Entering maxim
Entering maxim
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Entering maxim
Entering maxim
Entering maxim
colnames  4171
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colnames  4173
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enterinf minimum
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colnames  4176
Entering maxim
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colnames  4177
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colnames  4179
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colnames  4180
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colnames  4182
Entering maxim
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colnames  4183
Entering maxim
Entering maxim
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colnames  4184
enterinf minimum
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Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
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colnames  4185
enterinf minimum
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Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
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colnames  4186
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enterinf minimum
Entering maxim
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colnames  4189
Entering maxim
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colnames  4190
Entering maxim
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colnames  4191
Entering maxim
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colnames  4192
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Entering maxim
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colnames  4194
Entering maxim
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colnames  4195
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colnames  4196
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colnames  4197
Entering maxim
Entering maxim
Entering maxim
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colnames  4203
Entering maxim
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Entering maxim
Entering maxim
Entering maxim
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colnames  4205
Entering maxim
enterinf minimum
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Entering maxim
Entering maxim
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4206
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4207
Entering maxim
colnames  4208
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4209
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4210
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4211
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4212
colnames  4213
colnames  4214
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4215
colnames  4216
colnames  4217
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4218
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4219
colnames  4220
colnames  4221
colnames  4222
enterinf minimum
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4223
Entering maxim
Entering maxim
colnames  4224
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4225
colnames  4226
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4227
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4228
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4229
Entering maxim
colnames  4230
colnames  4231
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4232
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4233
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4234
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4235
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4236
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4237
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4238
colnames  4239
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4240
colnames  4241
Entering maxim
colnames  4242
colnames  4243
colnames  4244
colnames  4245
colnames  4246
colnames  4247
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4248
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4249
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4250
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4251
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4252
colnames  4253
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4254
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4255
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4256
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4257
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4258
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4259
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4260
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4261
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4262
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4263
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4264
colnames  4265
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4266
colnames  4267
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4268
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4269
colnames  4270
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4271
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4272
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4273
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4274
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4275
enterinf minimum
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4276
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4277
colnames  4278
colnames  4279
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4280
Entering maxim
Entering maxim
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
colnames  4281
colnames  4282
colnames  4283
colnames  4284
Entering maxim
colnames  4285
Entering maxim
Entering maxim
Entering maxim
colnames  4286
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
enterinf minimum
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
enterinf minimum
colnames  4287
colnames  4288
colnames  4289
colnames  4290
colnames  4291
colnames  4292
colnames  4293
colnames  4294
colnames  4295
colnames  4296
colnames  4297
colnames  4298
colnames  4299
colnames  4300
colnames  4301
colnames  4302
colnames  4303
colnames  4304
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4305
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4306
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4307
colnames  4308
colnames  4309
colnames  4310
colnames  4311
colnames  4312
colnames  4313
colnames  4314
colnames  4315
colnames  4316
colnames  4317
colnames  4318
colnames  4319
colnames  4320
colnames  4321
colnames  4322
colnames  4323
colnames  4324
colnames  4325
colnames  4326
colnames  4327
colnames  4328
colnames  4329
colnames  4330
colnames  4331
colnames  4332
colnames  4333
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4334
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4335
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
Entering maxim
colnames  4336
colnames  4337
colnames  4338
colnames  4339
colnames  4340
In [432]:
#updated dataframe after fixing outliers
new_data.head()
Out[432]:
week_num yield 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 ... 4311 4312 4313 4314 4315 4316 4317 4318 4319 4320 4321 4322 4323 4324 4325 4326 4327 4328 4329 4330 4331 4332 4333 4334 4335 4336 4337 4338 4339 4340
0 Week 48 FY15 0.57 0.765056 -0.356439 0.197052 0.795395 0 -0.001015 0.196354 0.081835 0.950103 -0.063364 -0.829207 -0.241069 -0.331018 -0.330876 0.018767 0.260456 -0.067674 0.525733 -0.131478 -0.474496 -0.229206 -0.436584 -0.391205 -0.291944 -0.394279 0.015729 -0.661840 -0.221699 ... -1.3 -1.036887 -0.991674 -1.021653 -0.991608 -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994 -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988 -0.980759 -0.730907 -0.99136 -0.774115 -0.724674 -0.542333 -0.992039 -1.036513 -0.981902 -0.995691 -0.800299
1 Week 49 FY15 0.47 1.062557 -0.156156 -0.234454 0.950919 0 -0.170481 -0.057973 0.362018 0.590900 -0.030287 -0.808753 0.520046 -0.331018 1.098410 -0.036730 1.044749 -0.154249 0.656448 -0.159455 -0.758297 -0.190549 -0.332570 0.349147 0.341943 -0.310972 0.303210 -0.558677 -0.133704 ... -1.3 -1.036887 -0.991674 -1.021653 -0.991608 -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994 -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988 -0.980759 -0.730907 -0.99136 -0.774115 -0.724674 -0.542333 -0.992039 -1.036513 -0.981902 -0.995691 -0.800299
2 Week 50 FY15 0.43 0.961968 -0.276802 0.106828 1.075338 0 -0.004001 0.110006 0.201414 0.516113 -0.016122 -0.772992 0.347968 -0.331018 0.772261 0.225179 0.792992 -0.104244 0.553572 -0.175071 -0.563206 0.008438 -0.469626 0.413091 0.442866 -0.420743 0.414846 -0.595874 -0.101545 ... -1.3 -1.036887 -0.991674 -1.021653 -0.991608 -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994 -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988 -0.980759 -0.730907 -0.99136 -0.774115 -0.724674 -0.542333 -0.992039 -1.036513 -0.981902 -0.995691 -0.800299
3 Week 51 FY15 0.49 0.864231 -0.411954 0.174596 1.168652 0 0.040849 0.032461 0.311033 0.605321 -0.023123 -0.711317 -0.098511 -0.331018 -0.485911 -0.064177 0.610803 -0.110098 0.553419 -0.201746 0.084896 -0.078933 -0.542146 0.656361 0.552064 -0.478825 0.474788 -0.449464 -0.085681 ... -1.3 -1.036887 -0.991674 -1.021653 -0.991608 -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994 -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988 -0.980759 -0.730907 -0.99136 -0.774115 -0.724674 -0.542333 -0.992039 -1.036513 -0.981902 -0.995691 -0.800299
4 Week 52 FY15 0.57 -0.083462 -0.212342 0.110023 1.251599 0 0.021135 -0.133111 0.152851 0.944587 -0.014222 -0.840133 0.974383 -0.331018 0.471113 -0.287214 0.642511 -0.140182 0.083886 -0.212156 -0.806272 -0.432690 -0.568484 0.453516 0.341360 -0.499920 0.375777 -0.458860 -0.107713 ... -1.3 -1.036887 -0.991674 -1.021653 -0.991608 -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994 -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988 -0.980759 -0.730907 -0.99136 -0.774115 -0.724674 -0.542333 -0.992039 -1.036513 -0.981902 -0.995691 -0.800299

5 rows × 4342 columns

In [433]:
# Statistics for each column in updated dataframe
new_data.describe()
Out[433]:
yield 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 ... 4311 4312 4313 4314 4315 4316 4317 4318 4319 4320 4321 4322 4323 4324 4325 4326 4327 4328 4329 4330 4331 4332 4333 4334 4335 4336 4337 4338 4339 4340
count 115.000000 131.000000 131.000000 131.000000 131.000000 131.0 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 1.310000e+02 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 ... 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000 131.000000
mean 0.675391 0.012175 -0.319763 0.032254 0.026639 0.0 -0.113118 0.052832 -0.019941 0.052197 -0.102723 -0.007656 -0.122928 -3.310184e-01 -0.008835 0.006114 0.008816 -0.071727 0.005120 -0.197256 -0.002396 0.132116 -0.140989 0.072317 0.094681 -0.175510 0.108733 -0.181117 -0.167583 -0.003357 ... 0.019847 -0.018212 0.015140 -0.012253 0.015139 0.013703 -0.013526 0.015146 0.013796 -0.017083 0.019840 -0.013002 0.019632 -0.017392 0.014745 0.015139 -0.010341 -0.023997 0.014198 0.014973 0.011159 0.015135 -0.041483 -0.050189 -0.181834 0.015146 -0.016099 0.014991 -0.016266 -0.018191
std 0.173095 0.992832 0.172818 0.974747 0.987683 0.0 0.194482 0.523170 0.296463 0.550490 0.034916 0.952971 0.473595 3.343455e-16 1.004106 0.272519 1.007099 0.249999 0.973037 0.030170 0.938170 0.408033 0.489089 0.440006 0.394875 0.362779 0.298533 0.471939 0.055647 0.943179 ... 0.998339 1.000419 1.003851 1.006448 1.003852 1.005233 1.005352 1.003845 1.005148 1.001739 0.998349 1.005716 0.998624 1.001403 1.004245 1.003852 1.007746 0.992102 1.004773 1.004018 1.007339 1.003856 0.912571 0.839993 0.465584 1.003845 1.002811 1.004001 1.002566 1.000447
min 0.380000 -2.276594 -0.600320 -1.344745 -1.765569 0.0 -0.428650 -1.146657 -0.723159 -1.156419 -0.186073 -0.885389 -1.071456 -3.310184e-01 -1.373330 -0.642826 -2.231267 -0.415026 -2.738574 -0.240134 -1.307903 -0.865598 -0.568484 -0.990475 -0.914732 -0.499920 -0.684218 -0.739742 -0.305234 -1.150176 ... -1.300000 -1.036887 -0.991674 -1.021653 -0.991608 -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994 -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988 -0.980759 -0.730907 -0.991360 -0.774115 -0.724674 -0.542333 -0.992039 -1.036513 -0.981902 -0.995691 -0.800299
25% 0.525000 -0.726656 -0.445459 -0.786582 -0.733926 0.0 -0.277096 -0.212699 -0.212299 -0.212930 -0.125539 -0.753347 -0.460810 -3.310184e-01 -0.739639 -0.160585 -0.418765 -0.286940 -0.653723 -0.220940 -0.637095 -0.097085 -0.544300 -0.175117 -0.156850 -0.474049 -0.070744 -0.549544 -0.202120 -0.805935 ... -1.300000 -1.036887 -0.991674 -1.021653 -0.991608 -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994 -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988 -0.980759 -0.730907 -0.991360 -0.774115 -0.724674 -0.542333 -0.992039 -1.036513 -0.981902 -0.995691 -0.800299
50% 0.650000 0.131757 -0.351201 -0.017179 0.162930 0.0 -0.149212 0.104851 -0.024938 0.185155 -0.110266 -0.408105 -0.229996 -3.310184e-01 -0.285495 -0.025708 0.314754 -0.104244 0.258382 -0.201746 -0.243295 0.129581 -0.334035 0.083081 0.126280 -0.310972 0.096966 -0.317995 -0.174289 -0.164899 ... 0.588668 0.894109 0.936147 0.457846 0.862280 -0.260324 0.470975 0.948911 -0.703088 0.537219 0.617484 -0.121098 0.582681 -0.811748 0.492216 0.934064 -0.752612 0.357856 -0.353018 0.534925 -0.730907 0.927847 -0.178247 -0.202833 -0.333432 0.927506 0.839520 0.742849 0.401413 -0.800299
75% 0.820000 0.868233 -0.249513 0.689194 0.930182 0.0 0.022078 0.409939 0.128273 0.416063 -0.081012 0.559834 0.113267 -3.310184e-01 0.653403 0.160909 0.789570 0.104917 0.736178 -0.179883 0.497446 0.415257 0.155760 0.368455 0.348405 0.045178 0.338238 0.054899 -0.133377 0.484090 ... 0.842512 0.941332 0.990198 1.010513 1.013364 0.861445 1.009165 1.001910 1.148301 0.961256 0.861348 1.022361 0.879483 1.184753 0.932457 0.986209 0.903354 0.870551 0.958373 1.049698 1.022801 0.985124 0.253157 0.328903 0.009021 1.000359 0.960832 0.950500 0.838910 1.095020
max 1.070000 1.711989 0.044404 2.743958 1.251599 0.0 0.470839 1.142294 0.639133 1.359553 -0.014222 2.529605 0.974383 -3.310184e-01 2.742967 0.643150 1.312071 0.692702 2.087265 -0.118297 2.199259 0.873747 1.205849 1.102200 1.056587 0.824019 0.828886 0.961564 -0.030263 2.419128 ... 0.995714 1.200532 1.182414 1.316671 1.102186 2.091451 1.323212 1.125495 1.559355 1.204925 0.968651 1.527287 1.077952 1.405106 1.871936 1.188678 2.651805 1.949017 1.764962 1.386134 1.936779 1.223061 1.794064 1.909268 0.836052 1.187765 1.114567 1.424802 1.683484 1.713457

8 rows × 4341 columns

In [434]:
# Histogram Plot of variable 1
figsize(8, 8)
plt.hist(data['1'].dropna(), bins = 20, edgecolor = 'black');
plt.xlabel('1'); 
plt.ylabel('Count'); plt.title('Variable 1 Distribution');

This plot looks close to normally distributed with a long tail on the left side (it has a negative skew).

In [435]:
figsize(8, 8)
# Histogram of the yield
plt.style.use('fivethirtyeight')
plt.hist(new_data['yield'].dropna(), bins = 100, edgecolor = 'k');
plt.xlabel('yield'); plt.ylabel('Number of Yelds'); 
plt.title('Yield Distribution');

The yield label seems also to have a normal distribution.

Looking for Relationships

Since there are no categorical values to plot against the yield variable's values on a kernel density estimation plot, I will to focus on the numerical columns for predicting the yield.

Correlations between Features and Target

In order to quantify correlations between the features (variables) and the target (yield variable), we can calculate the Pearson correlation coefficient. This is a measure of the strength and direction of a linear relationship between two variables: a value of -1 means the two variables are perfectly negatively linearly correlated and a value of +1 means the two variables are perfectly positively linearly correlated.

In [436]:
# Find all correlations and sort 
correlations_data = new_data.corr()['yield'].sort_values()

# Print the most negative correlations
print(correlations_data.head(15), '\n')

# Print the most positive correlations
print(correlations_data.tail(15))
4      -0.803702
4126   -0.792725
1956   -0.792725
314    -0.789987
2484   -0.789987
3816   -0.788481
1646   -0.788481
3506   -0.783567
1336   -0.783567
4100   -0.776829
1930   -0.776829
2794   -0.771485
624    -0.771485
3196   -0.766431
1026   -0.766431
Name: yield, dtype: float64 

5      NaN
315    NaN
625    NaN
935    NaN
1245   NaN
1555   NaN
1865   NaN
2175   NaN
2485   NaN
2512   NaN
2795   NaN
3105   NaN
3415   NaN
3725   NaN
4035   NaN
Name: yield, dtype: float64

Correlation is calculated as folows : $cor(i,j) = cov(i,j)/[stdev(i)*stdev(j)]$

If the values of the ith or jth variable do not vary, then the respective standard deviation will be zero and so will the denominator of the fraction. Thus, the correlation will be NaN. If there is no correlation, then in my opinion, the variables in cause should not be used in predicting the yield values.

In [437]:
#drop the columns with constant value of zero
new_data.drop(['5','315','625','935','1245','1555','1865','2175','2485','2795','3105','3415','3725','4035'], axis=1, inplace=True)
In [438]:
new_data.head()
Out[438]:
week_num yield 1 2 3 4 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 ... 4311 4312 4313 4314 4315 4316 4317 4318 4319 4320 4321 4322 4323 4324 4325 4326 4327 4328 4329 4330 4331 4332 4333 4334 4335 4336 4337 4338 4339 4340
0 Week 48 FY15 0.57 0.765056 -0.356439 0.197052 0.795395 -0.001015 0.196354 0.081835 0.950103 -0.063364 -0.829207 -0.241069 -0.331018 -0.330876 0.018767 0.260456 -0.067674 0.525733 -0.131478 -0.474496 -0.229206 -0.436584 -0.391205 -0.291944 -0.394279 0.015729 -0.661840 -0.221699 -1.141306 ... -1.3 -1.036887 -0.991674 -1.021653 -0.991608 -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994 -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988 -0.980759 -0.730907 -0.99136 -0.774115 -0.724674 -0.542333 -0.992039 -1.036513 -0.981902 -0.995691 -0.800299
1 Week 49 FY15 0.47 1.062557 -0.156156 -0.234454 0.950919 -0.170481 -0.057973 0.362018 0.590900 -0.030287 -0.808753 0.520046 -0.331018 1.098410 -0.036730 1.044749 -0.154249 0.656448 -0.159455 -0.758297 -0.190549 -0.332570 0.349147 0.341943 -0.310972 0.303210 -0.558677 -0.133704 -1.096242 ... -1.3 -1.036887 -0.991674 -1.021653 -0.991608 -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994 -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988 -0.980759 -0.730907 -0.99136 -0.774115 -0.724674 -0.542333 -0.992039 -1.036513 -0.981902 -0.995691 -0.800299
2 Week 50 FY15 0.43 0.961968 -0.276802 0.106828 1.075338 -0.004001 0.110006 0.201414 0.516113 -0.016122 -0.772992 0.347968 -0.331018 0.772261 0.225179 0.792992 -0.104244 0.553572 -0.175071 -0.563206 0.008438 -0.469626 0.413091 0.442866 -0.420743 0.414846 -0.595874 -0.101545 -1.017314 ... -1.3 -1.036887 -0.991674 -1.021653 -0.991608 -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994 -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988 -0.980759 -0.730907 -0.99136 -0.774115 -0.724674 -0.542333 -0.992039 -1.036513 -0.981902 -0.995691 -0.800299
3 Week 51 FY15 0.49 0.864231 -0.411954 0.174596 1.168652 0.040849 0.032461 0.311033 0.605321 -0.023123 -0.711317 -0.098511 -0.331018 -0.485911 -0.064177 0.610803 -0.110098 0.553419 -0.201746 0.084896 -0.078933 -0.542146 0.656361 0.552064 -0.478825 0.474788 -0.449464 -0.085681 -0.889290 ... -1.3 -1.036887 -0.991674 -1.021653 -0.991608 -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994 -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988 -0.980759 -0.730907 -0.99136 -0.774115 -0.724674 -0.542333 -0.992039 -1.036513 -0.981902 -0.995691 -0.800299
4 Week 52 FY15 0.57 -0.083462 -0.212342 0.110023 1.251599 0.021135 -0.133111 0.152851 0.944587 -0.014222 -0.840133 0.974383 -0.331018 0.471113 -0.287214 0.642511 -0.140182 0.083886 -0.212156 -0.806272 -0.432690 -0.568484 0.453516 0.341360 -0.499920 0.375777 -0.458860 -0.107713 -1.150176 ... -1.3 -1.036887 -0.991674 -1.021653 -0.991608 -0.897575 -1.020315 -0.992045 -0.903636 -1.028505 -1.299491 -0.976994 -1.285899 -0.811748 -0.965811 -0.991597 -0.752612 -1.001958 -0.929988 -0.980759 -0.730907 -0.99136 -0.774115 -0.724674 -0.542333 -0.992039 -1.036513 -0.981902 -0.995691 -0.800299

5 rows × 4328 columns

In [439]:
#correlation coeficient for data without the variables with constant value
# Find all correlations 
correlations_data = new_data.corr()['yield'].sort_values()

# Print the most negative correlations
print(correlations_data.head(15), '\n')

# Print the most positive correlations
print(correlations_data.tail(30))
4      -0.803702
4126   -0.792725
1956   -0.792725
314    -0.789987
2484   -0.789987
3816   -0.788481
1646   -0.788481
3506   -0.783567
1336   -0.783567
4100   -0.776829
1930   -0.776829
2794   -0.771485
624    -0.771485
3196   -0.766431
1026   -0.766431
Name: yield, dtype: float64 

2801     0.718458
2819     0.718631
631      0.718646
4047     0.724399
1877     0.724399
1863     0.731834
4033     0.731834
3723     0.735511
1553     0.735511
1567     0.735911
3737     0.735911
1243     0.740894
3413     0.740894
2483     0.745065
313      0.745065
1257     0.747874
3427     0.747874
3103     0.754794
933      0.754794
623      0.756456
2793     0.756456
17       0.757064
947      0.759665
3117     0.759665
2807     0.768588
637      0.768588
2497     0.773458
327      0.773458
yield    1.000000
2512          NaN
Name: yield, dtype: float64

To account for possible non-linear relationships, we can take square root and natural log transformations of the features and then calculate the correlation coefficients with the yield. Bellow there is the function to performs the variables' square root and log transformation; in the case of this data set where there are many variables, the process is resourcess consuming and I won't perfom it.

In [440]:
# Select the numeric columns
#numeric_subset = new_data.select_dtypes('number')

# Create columns with square root and log of numeric columns
#for col in numeric_subset.columns:
     #Skip the yield column
    #if col == ['yield','week_num']:
        #next
    #else:
        #numeric_subset['sqrt_' + str(col)] = np.sqrt(numeric_subset[col])
        #numeric_subset['log_' + str(col)] = np.log(numeric_subset[col])

# Drop buildings without an energy star score
#features = numeric_subset.dropna(subset = ['yield'])

# Find correlations with the score 
#correlations = features.corr()['yeld'].dropna().sort_values()
In [441]:
#save the data set with a new name
features=new_data.copy()

Pairs Plot

As a final step for exploratory data analysis, we can make a pairs plot between several different variables. The Pairs Plot is a great way to examine many variables at once as it shows scatterplots between pairs of variables and histograms of single variables on the diagonal.

Using the seaborn PairGrid function, we can map different plots on to the three aspects of the grid. The upper triangle will have scatterplots, the diagonal will show histograms, and the lower triangle will show both the correlation coefficient between two variables and a 2-D kernel density estimate of the two variables.

In [442]:
# Extract the columns to  plot
plot_data = features[['yield', '4', 
                      '1956', '4126']]

# Replace the inf with nan
plot_data = plot_data.replace({np.inf: np.nan, -np.inf: np.nan})

# Drop na values
plot_data = plot_data.dropna()

# Function to calculate correlation coefficient between two columns
def corr_func(x, y, **kwargs):
    r = np.corrcoef(x, y)[0][1]
    ax = plt.gca()
    ax.annotate("r = {:.2f}".format(r),
                xy=(.2, .8), xycoords=ax.transAxes,
                size = 20)

# Create the pairgrid object
grid = sns.PairGrid(data = plot_data, size = 3)

# Upper is a scatter plot
grid.map_upper(plt.scatter, color = 'red', alpha = 0.6)

# Diagonal is a histogram
grid.map_diag(plt.hist, color = 'red', edgecolor = 'black')

# Bottom is correlation and density plot
grid.map_lower(corr_func);
grid.map_lower(sns.kdeplot, cmap = plt.cm.Reds)

# Title for entire plot
plt.suptitle('Pairs Plot of Hypothetical Weather Data', size = 36, y = 1.02);
C:\Users\Daniela\Anaconda3.5\lib\site-packages\seaborn\axisgrid.py:1241: UserWarning: The `size` paramter has been renamed to `height`; please update your code.
  warnings.warn(UserWarning(msg))
C:\Users\Daniela\Anaconda3.5\lib\site-packages\scipy\stats\stats.py:1713: FutureWarning: Using a non-tuple sequence for multidimensional indexing is deprecated; use `arr[tuple(seq)]` instead of `arr[seq]`. In the future this will be interpreted as an array index, `arr[np.array(seq)]`, which will result either in an error or a different result.
  return np.add.reduce(sorted[indexer] * weights, axis=axis) / sumval

To interpret the relationships in the plot, we can look for where the variables in one row intersect with the variables in one column. For example, to find the relationship between yield and 4126, we look at the yield column and find 4126 row. At the intersection (the lower left plot) we see that the score has a -0.79 correlation coefficient with this varible. If we look at the upper right plot, we can see a scatterplot of this relationship.

Feature Engineering and Selection

For feature selection, I will remove the collinear features. I will discuss collinearity (also called multicollinearity) when I get to that process!

First will plot the collinear feature into a heat map for inspection.

In [443]:
sns.heatmap(features.corr(), cmap="coolwarm")
Out[443]:
<matplotlib.axes._subplots.AxesSubplot at 0x288ba281a58>

It seems that the correlation coefficient has maximum value around 0.8 (both positive and negative).

Removing Collinear Features

Highly collinear features have a significant correlation coefficent between them. Having highly correlated fetures (positive and negative) can impact the performance of model by multicollinearity.

This can lead to misleading results, but luckily decision trees and boosted trees algorithms are immune to multicollinearity by nature.

Here I will remove the features that have a colinearity coefficient of 0.75 (which is close to the maximum coeficient, but many of the features have coefficient around 0.8). Setting up a higher or lower threshold won't help impove much in algorithm performance. As mentioned above, for this specific problem removing colinear feature is not necessary, as in the end I will be using a boosted tree type algortihm; it mostly helps in decreasing resources consume, as we will end up with less features to feed the algorithm.

In [444]:
def remove_collinear_features(x, threshold):
    '''
    Objective:
        Remove collinear features in a dataframe with a correlation coefficient
        greater than the threshold. Removing collinear features can help a model
        to generalize and improves the interpretability of the model.
        
    Inputs: 
        threshold: any features with correlations greater than this value are removed
    
    Output: 
        dataframe that contains only the non-highly-collinear features
    '''
    
    # Dont want to remove correlations between Energy Star Score
    y = x['yield']
    x = x.drop(columns = ['yield'])
    
    # Calculate the correlation matrix
    corr_matrix = x.corr()
    iters = range(len(corr_matrix.columns) - 1)
    drop_cols = []

    # Iterate through the correlation matrix and compare correlations
    for i in iters:
        for j in range(i):
            item = corr_matrix.iloc[j:(j+1), (i+1):(i+2)]
            col = item.columns
            row = item.index
            val = abs(item.values)
            
            # If correlation exceeds the threshold
            if val >= threshold:
                # Print the correlated features and the correlation value
                # print(col.values[0], "|", row.values[0], "|", round(val[0][0], 2))
                drop_cols.append(col.values[0])

    # Drop one of each pair of correlated columns
    drops = set(drop_cols)
    x = x.drop(columns = drops)
    
    
    # Add the score back in to the data
    x['yield'] = y
               
    return x
In [445]:
# Remove the collinear features above a specified correlation coefficient
features = remove_collinear_features(features, 0.75);
C:\Users\Daniela\Anaconda3.5\lib\site-packages\ipykernel_launcher.py:33: RuntimeWarning: invalid value encountered in greater_equal

Additional Feature Selection

There are plenty of more methods for feature selection. Some popular methods include principal components analysis (PCA) which transforms the features into a reduced number of dimensions that preserve the greatest variance, or independent components analysis (ICA) which aims to find the independent sources in a set of features. However, while these methods are effective at reducing the number of features, they create new features that have no physical meaning and hence make interpreting a model nearly impossible.

In [446]:
features.to_csv('features.csv', index = False)
features = pd.read_csv('features.csv')
features.shape
Out[446]:
(131, 554)

Split Into Training and Testing Sets

In machine learning, we always need to separate our features into two sets:

  • Training set which we provide to our model during training along with the answers so it can learn a mapping between the features and the target; I will split again this set into a train and validation set, as the test set is not containing the target values column. Since we are not dealling with classification and class imbalance, we do not need to do stratified subsampling.
  • Testing set which we use to evaluate the mapping learned by the model. The model has never seen the answers on the testing set, but instead, must make predictions using only the features.

Splitting the data into a random training and validation set is simple using scikit-learn. We can set the random state of the split to ensure consistent results.

In [447]:
#split the data set into training and testing set as was indicated in the problem statement
train = features.iloc[0:115].copy()
test = features.iloc[115:131].copy()
#test = test.drop(columns='yield')

# Display sizes of data
print('Training Feature Size: ', train.shape)
print('Testing Feature Size:  ', test.shape)
Training Feature Size:  (115, 554)
Testing Feature Size:   (16, 554)
In [448]:
#save the training and testing set into .csv files
train.to_csv('train_final', index=False)
test.to_csv('test_final', index=False)
In [449]:
#save the target column and convert y to one-dimensional array (vector)
y =np.array(train['yield'])
# Remove the ids and target and convert X to  array (vector)
X =np.array (train.drop(columns = ['week_num', 'yield']))
In [450]:
def mape(y_true, y_pred):
    return (np.mean(abs((y_true - y_pred)/y_true)))*100

Models to Evaluate

We will compare five different machine learning models using the great Scikit-Learn library:

  1. Base-line model which is a naive gues based on median values
  2. Linear Regression
  3. Random Forest Regression
  4. Light Gradient Boosting Regression
  5. Gradient Boosting Regression
  6. K-Nearest Neighbors Regression
  7. Support Vector Machine Regression
In [451]:
from sklearn.model_selection import train_test_split
x_train, x_valid, y_train, y_valid = train_test_split(X, y, test_size=0.2, random_state=42)
    
baseline_guess = np.median(y)
err = mape(y_valid, baseline_guess)
print('MAPE = ' + str(err))

MAPE_baseline = err
MAPE = 21.51906573076165
In [452]:
# Takes in a model, trains the model, and evaluates the model on the validation set
def fit_and_evaluate(model):

    x_train, x_valid, y_train, y_valid = train_test_split(X, y, test_size=0.2, random_state=42)
    
    # Train the model
    model.fit(x_train, y_train)
    
     # Make predictions and evalute
    preds = model.predict(x_valid)
    err = mape(y_valid, preds)
    print('MAPE = ' + str(err))
    
    MAPE=err
    # Return the performance metric list
    return  MAPE
In [453]:
lr = LinearRegression()
lr_mape = fit_and_evaluate(lr)
MAPE = 16.15309448780313
In [454]:
random_forest = RandomForestRegressor(random_state=60)
random_forest_mape = fit_and_evaluate(random_forest)
MAPE = 8.99545317633642
In [455]:
knn = KNeighborsRegressor(n_neighbors=5)
knn_mape = fit_and_evaluate(knn)
MAPE = 15.66753359085877
In [456]:
gradient_boosted = GradientBoostingRegressor()
gradient_boosted_mape = fit_and_evaluate(gradient_boosted)
MAPE = 8.603293440600064
In [457]:
lgbm = lgb.LGBMRegressor()
lgbm_mape = fit_and_evaluate(lgbm)
MAPE = 8.78521446433091
In [458]:
svm = SVR(C = 1000, gamma = 0.1)
svm_mape = fit_and_evaluate(svm)

print('Support Vector Machine Regression Performance on the test set: MAE = %0.4f' % svm_mape)
MAPE = 26.660652256377805
Support Vector Machine Regression Performance on the test set: MAE = 26.6607
In [459]:
plt.style.use('fivethirtyeight')
figsize(8, 6)

# Dataframe to hold the results
model_comparison = pd.DataFrame({'model': ['Linear Regression', 
                                           'Random Forest', 'Light Gradient Boosted Machine', 'Gradient Boosted Machine',
                                            'K-Nearest Neighbors', 'Support Vector Machine', 'Baseline'],
                                 'mape': [lr_mape, random_forest_mape, 
                                         lgbm_mape,gradient_boosted_mape, knn_mape, svm_mape,MAPE_baseline]})

# Horizontal bar chart of test mae
model_comparison.sort_values('mape', ascending = False).plot(x = 'model', y = 'mape', kind = 'barh',
                                                           color = 'red', edgecolor = 'black')

# Plot formatting
plt.ylabel(''); plt.yticks(size = 14); plt.xlabel('Mean Absolute Percentage Error'); plt.xticks(size = 14)
plt.title('Model Comparison on validation MAPE', size = 20);

Depending on the run (the exact results change slighty each time), the Gradient Boosting Regressor performs the best followed by Light Gradient Boosting Regressor and the random forrest regressor. I have to admit that this is not the most fair comparison because we are using mostly the default hyperparameters. Especially with the Support Vector Regressor, the hyperparameters have a significant influence on performance. (the random forest and gradient boosting methods are great for starting out because the performance is less dependent on the model settings). I will continue working with gradient boosting regressor (and not random forrest) and optimizing the best model using hyperparamter tuning. Here are the reasons:

  • RF (Random Forest) uses decision trees, which are very prone to overfitting. In order to achieve higher accuracy, RF decides to create a large number of them based on bagging. The basic idea is to resample the data over and over and for each sample train a new classifier. Different classifiers overfit the data in a different way, and through voting those differences are averaged out.
  • GBM is a boosting method, which builds on weak classifiers. The idea is to add a classifier at a time, so that the next classifier is trained to improve the already trained ensemble ( selection of sample is made more intelligently, subsequently giving more and more weight to hard to classify observations). Notice that for RF each iteration the classifier is trained independently from the rest.

Model Optimization

In machine learning, optimizing a model means finding the best set of hyperparameters for a particular problem.

Hyperparameters

First, to understand what model hyperparameters are in contrast to model parameters :

  • Model hyperparameters are best thought of as settings for a machine learning algorithm that are tuned by the data scientist before training. Examples would be the number of trees in the random forest, or the number of neighbors used in K Nearest Neighbors Regression.
  • Model parameters are what the model learns during training, such as the weights in the linear regression.

We control a model by choosing the hyperparameters, and these choices can have a significant effect on the final performance of the model (although usually not as great of an effect as getting more data or engineering features).

Tuning the model hyperparameters controls the balance of under vs over fitting in a model. We can try to correct for under-fitting by making a more complex model, such as using more trees in a random forest. A model that underfits has high bias, and occurs when our model does not have enough capacity (degrees of freedom) to learn the relationship between the features and the target. We can try to correct for overfitting by limiting the complexity of the model and applying regularization. This might mean decreasing the degree of a polynomial regression.. A model that overfits has high variance and in effect has memorized the training set. Both underfitting and overfitting lead to poor generalization performance on the test set.

The problem with choosing the hyperparameters is that no set will work best across all problems. Therefore, for every new dataset, we have to find the best settings. This can be a time-consuming process, but luckily there are several options for performing this procedure in Scikit-Learn. Even better, new libraries, such as TPOT and HyperOpt (Bayesian Hyperparameter Optimization), is aiming to do this process automatically ! For now, I will stick to doing this manually (sort of) in Scikit-Learn!

Hyperparameter Tuning with Random Search and Cross Validation

We can choose the best hyperparameters for a model through random search and cross validation.

  • Random search refers to the method in which we choose hyperparameters to evaluate: we define a range of options, and then randomly select combinations to try. This is in contrast to grid search which evaluates every single combination we specify. Generally, random search is better when we have limited knowledge of the best model hyperparameters and we can use random search to narrow down the options and then use grid search with a more limited range of options.
  • Cross validation is the method used to assess the performance of the hyperparameters. Rather than splitting the training set up into separate training and validation sets which reduces the amount of training data we can use, we use K-Fold Cross Validation. This means dividing the training data into K folds, and then going through an iterative process where we first train on K-1 of the folds and then evaluate performance on the kth fold. We repeat this process K times so eventually we will have tested on every example in the training data with the key that each iteration we are testing on data that we did not train on. At the end of K-fold cross validation, we take the average error on each of the K iterations as the final performance measure and then train the model on all the training data at once. The performance we record is then used to compare different combinations of hyperparameters.

Here I will implement random search with cross validation to select the optimal hyperparameters for the gradient boosting regressor. I will first define a grid then peform an iterative process of: randomly sample a set of hyperparameters from the grid, evaluate the hyperparameters using 4-fold cross-validation, and then select the hyperparameters with the best performance.

In [460]:
# Loss function to be optimized
loss = ['ls']
#loss = ['ls', 'lad', 'huber']
# Number of trees used in the boosting process
n_estimators = [100, 500, 900, 1100, 1500]

# Maximum depth of each tree
max_depth = [2, 3, 5, 10, 15]

# Minimum number of samples per leaf
min_samples_leaf = [1, 2, 4, 6, 8]

# Minimum number of samples to split a node
min_samples_split = [2, 4, 6, 10, 12]

# Maximum number of features to consider for making splits
max_features = ['auto', 'sqrt', 'log2', None]

# Define the grid of hyperparameters to search
hyperparameter_grid = {'loss': loss,
                       'n_estimators': n_estimators,
                       'max_depth': max_depth,
                       'min_samples_leaf': min_samples_leaf,
                       'min_samples_split': min_samples_split,
                       'max_features': max_features}
In [461]:
#Split the training set into train and validation set
x_train, x_valid, y_train, y_valid = train_test_split(X, y, test_size=0.2, random_state = 42)
model =GradientBoostingRegressor()

# Set up the random search with 4-fold cross validation
random_cv = RandomizedSearchCV(estimator=model,
                               param_distributions=hyperparameter_grid,
                               cv=4, n_iter=25, 
                               scoring = 'neg_mean_absolute_error',
                               n_jobs = -1, verbose = 1, 
                               return_train_score = True,
                               random_state=42)
random_cv.fit(x_train, y_train)
Fitting 4 folds for each of 25 candidates, totalling 100 fits
[Parallel(n_jobs=-1)]: Done  42 tasks      | elapsed:   14.5s
[Parallel(n_jobs=-1)]: Done 100 out of 100 | elapsed:   20.1s finished
Out[461]:
RandomizedSearchCV(cv=4, error_score='raise',
          estimator=GradientBoostingRegressor(alpha=0.9, criterion='friedman_mse', init=None,
             learning_rate=0.1, loss='ls', max_depth=3, max_features=None,
             max_leaf_nodes=None, min_impurity_decrease=0.0,
             min_impurity_split=None, min_samples_leaf=1,
             min_samples_split=2, min_weight_fraction_leaf=0.0,
             n_estimators=100, presort='auto', random_state=None,
             subsample=1.0, verbose=0, warm_start=False),
          fit_params=None, iid=True, n_iter=25, n_jobs=-1,
          param_distributions={'loss': ['ls'], 'n_estimators': [100, 500, 900, 1100, 1500], 'max_depth': [2, 3, 5, 10, 15], 'min_samples_leaf': [1, 2, 4, 6, 8], 'min_samples_split': [2, 4, 6, 10, 12], 'max_features': ['auto', 'sqrt', 'log2', None]},
          pre_dispatch='2*n_jobs', random_state=42, refit=True,
          return_train_score=True, scoring='neg_mean_absolute_error',
          verbose=1)
In [462]:
# Get all of the cv results and sort by the test performance
random_results = pd.DataFrame(random_cv.cv_results_).sort_values('mean_test_score', ascending = False)

random_results.head(3)
Out[462]:
mean_fit_time std_fit_time mean_score_time std_score_time param_n_estimators param_min_samples_split param_min_samples_leaf param_max_features param_max_depth param_loss params split0_test_score split1_test_score split2_test_score split3_test_score mean_test_score std_test_score rank_test_score split0_train_score split1_train_score split2_train_score split3_train_score mean_train_score std_train_score
11 0.039743 0.002046 0.0005 5.001428e-04 100 4 1 sqrt 2 ls {'n_estimators': 100, 'min_samples_split': 4, ... -0.054306 -0.057804 -0.084480 -0.071910 -0.067125 0.011993 1 -0.004404 -0.004081 -0.004255 -0.005858 -0.004650 0.000707
24 0.202966 0.005338 0.0010 4.578320e-07 900 12 8 sqrt 3 ls {'n_estimators': 900, 'min_samples_split': 12,... -0.071192 -0.062028 -0.074844 -0.069855 -0.069480 0.004674 2 -0.000178 -0.000170 -0.000173 -0.000200 -0.000180 0.000012
5 0.314700 0.003960 0.0010 3.769729e-07 1500 10 2 sqrt 15 ls {'n_estimators': 1500, 'min_samples_split': 10... -0.066434 -0.053679 -0.078538 -0.080963 -0.069903 0.010864 3 -0.000226 -0.000232 -0.000217 -0.000225 -0.000225 0.000005
In [463]:
random_cv.best_estimator_
Out[463]:
GradientBoostingRegressor(alpha=0.9, criterion='friedman_mse', init=None,
             learning_rate=0.1, loss='ls', max_depth=2,
             max_features='sqrt', max_leaf_nodes=None,
             min_impurity_decrease=0.0, min_impurity_split=None,
             min_samples_leaf=1, min_samples_split=4,
             min_weight_fraction_leaf=0.0, n_estimators=100,
             presort='auto', random_state=None, subsample=1.0, verbose=0,
             warm_start=False)

The best gradient boosted model has the following hyperparameters:

  • loss = ls
  • n_estimators = 100
  • max_depth = 10
  • min_samples_leaf = 4
  • min_samples_split = 6
  • max_features = 'None'

Using random search is a good method to narrow down the possible hyperparameters to try. Initially, we had no idea which combination would work the best, but this at least narrows down the range of options.

In [464]:
trees_grid = {'n_estimators': [100, 150, 200, 250, 300, 350, 400, 450, 500, 550, 600, 650, 700, 750, 800]}

model = GradientBoostingRegressor(loss = 'ls', max_depth = 2,
                                  min_samples_leaf = 4,
                                  min_samples_split = 6,
                                  max_features =None,
                                  n_estimators=100,
                                  random_state = 42)
grid_search = GridSearchCV(estimator = model, param_grid=trees_grid, cv = 4, 
                           scoring = 'neg_mean_absolute_error', verbose = 1,
                           n_jobs = -1, return_train_score = True)
grid_search.fit(x_train, y_train)
    
    
Fitting 4 folds for each of 15 candidates, totalling 60 fits
[Parallel(n_jobs=-1)]: Done  42 tasks      | elapsed:    7.7s
[Parallel(n_jobs=-1)]: Done  60 out of  60 | elapsed:   10.6s finished
Out[464]:
GridSearchCV(cv=4, error_score='raise',
       estimator=GradientBoostingRegressor(alpha=0.9, criterion='friedman_mse', init=None,
             learning_rate=0.1, loss='ls', max_depth=2, max_features=None,
             max_leaf_nodes=None, min_impurity_decrease=0.0,
             min_impurity_split=None, min_samples_leaf=4,
             min_samples_split=6, min_weight_fraction_leaf=0.0,
             n_estimators=100, presort='auto', random_state=42,
             subsample=1.0, verbose=0, warm_start=False),
       fit_params=None, iid=True, n_jobs=-1,
       param_grid={'n_estimators': [100, 150, 200, 250, 300, 350, 400, 450, 500, 550, 600, 650, 700, 750, 800]},
       pre_dispatch='2*n_jobs', refit=True, return_train_score=True,
       scoring='neg_mean_absolute_error', verbose=1)
In [465]:
results = pd.DataFrame(grid_search.cv_results_)
results.head()
Out[465]:
mean_fit_time std_fit_time mean_score_time std_score_time param_n_estimators params split0_test_score split1_test_score split2_test_score split3_test_score mean_test_score std_test_score rank_test_score split0_train_score split1_train_score split2_train_score split3_train_score mean_train_score std_train_score
0 0.262958 0.080836 0.000250 0.000433 100 {'n_estimators': 100} -0.078920 -0.061551 -0.071311 -0.078694 -0.072619 0.007085 15 -0.002180 -0.002165 -0.001703 -0.002508 -0.002139 0.000287
1 0.552162 0.044587 0.001251 0.000434 150 {'n_estimators': 150} -0.078552 -0.061534 -0.071282 -0.078204 -0.072393 0.006907 14 -0.000521 -0.000492 -0.000364 -0.000561 -0.000485 0.000074
2 0.678394 0.065337 0.001000 0.001225 200 {'n_estimators': 200} -0.078586 -0.061491 -0.071260 -0.078161 -0.072375 0.006924 1 -0.000249 -0.000255 -0.000255 -0.000260 -0.000255 0.000004
3 0.632900 0.047653 0.000500 0.000500 250 {'n_estimators': 250} -0.078586 -0.061491 -0.071260 -0.078161 -0.072375 0.006924 1 -0.000249 -0.000255 -0.000255 -0.000260 -0.000255 0.000004
4 0.545914 0.020159 0.000750 0.000433 300 {'n_estimators': 300} -0.078586 -0.061491 -0.071260 -0.078161 -0.072375 0.006924 1 -0.000249 -0.000255 -0.000255 -0.000260 -0.000255 0.000004
In [466]:
# Default model
default_model = GradientBoostingRegressor(random_state = 42)

# Select the best model
final_model = grid_search.best_estimator_

final_model
Out[466]:
GradientBoostingRegressor(alpha=0.9, criterion='friedman_mse', init=None,
             learning_rate=0.1, loss='ls', max_depth=2, max_features=None,
             max_leaf_nodes=None, min_impurity_decrease=0.0,
             min_impurity_split=None, min_samples_leaf=4,
             min_samples_split=6, min_weight_fraction_leaf=0.0,
             n_estimators=200, presort='auto', random_state=42,
             subsample=1.0, verbose=0, warm_start=False)
In [467]:
%%timeit -n 1 -r 5
default_model.fit(x_train, y_train)
413 ms ± 20 ms per loop (mean ± std. dev. of 5 runs, 1 loop each)
In [468]:
%%timeit -n 1 -r 5
final_model.fit(x_train, y_train)
488 ms ± 41.5 ms per loop (mean ± std. dev. of 5 runs, 1 loop each)
In [469]:
default_pred = default_model.predict(x_valid)
final_pred = final_model.predict(x_valid)

print('Default model performance on the validation set: MAPE = %0.4f.' % mape(y_valid, default_pred))
print('Final model performance on the validation set:   MAPE = %0.4f.' % mape(y_valid, final_pred))
Default model performance on the validation set: MAPE = 8.5393.
Final model performance on the validation set:   MAPE = 7.4867.

The final model does out-perform the baseline model by about 10%, and at the cost of very slightly increased running time. Machine learning is often a field of tradeoffs: bias vs variance, acccuracy vs interpretability, accuracy vs running time, and the final decision of which model to use depends on the situation. Here, the increase in run time is not an impediment; in a different situation, the balance might not be the same so we would need to consider what we are optimizing for and the limitations we have to work with.

To get a sense of the predictions, we can plot the distribution of true values on the validation set and the predicted values on the validation set.

In [470]:
figsize(8, 8)

# Density plot of the final predictions and the test values
sns.kdeplot(final_pred, label = 'Predictions')
sns.kdeplot(y_valid, label = 'Values')

# Label the plot
plt.xlabel('Energy Star Score'); plt.ylabel('Density');
plt.title('Validation Values and Predictions');
C:\Users\Daniela\Anaconda3.5\lib\site-packages\scipy\stats\stats.py:1713: FutureWarning: Using a non-tuple sequence for multidimensional indexing is deprecated; use `arr[tuple(seq)]` instead of `arr[seq]`. In the future this will be interpreted as an array index, `arr[np.array(seq)]`, which will result either in an error or a different result.
  return np.add.reduce(sorted[indexer] * weights, axis=axis) / sumval

The distribution looks to be nearly the same although the density of the predicted values is closer to the median of the validation values rather than to the actual peak at 0.6 and 1.0. It appears the model might be less accurate at predicting the extreme values and instead predicts values closer to the median.

Another diagnostic plot is a histogram of the residuals. Ideally, we would hope that the residuals are normally distributed, meaning that the model is wrong the same amount in both directions (high and low).

In [471]:
figsize = (6, 6)

# Calculate the residuals 
residuals = final_pred - y_valid

# Plot the residuals in a histogram
plt.hist(residuals, color = 'red', bins = 20,
         edgecolor = 'black')
plt.xlabel('Error'); plt.ylabel('Count')
plt.title('Distribution of Residuals');

The residuals are close to normally disributed, with a few noticeable outliers on the low end. These indicate errors where the model estimate was below that of the true value.

Preparing the testing set and make prediction

In [472]:
#save the ids for test set 
ids_test = test['week_num']
# Remove the ids and target
X_test = test.drop(columns = ['week_num', 'yield'])
X_test = np.array(X_test)
In [473]:
#make prediction on the test set
preds = final_model.predict(X_test)
In [474]:
# Prepare submission
subm = pd.DataFrame()
subm['week_num'] = ids_test.values
subm['yield'] = preds
subm.to_csv('submission_gbm.csv', index=False)

Feature Importances

One of the basic ways we can interpret an ensemble of decision trees is through what are known as the feature importances. These can be interpreted as the variables which are most predictive of the target. While the actual details of the feature importances are quite complex (here is a Stack Overflow question on the subject), we can use the relative values to compare the features and determine which are most relevant to our problem.

In [475]:
train_features = train.drop(columns = ['week_num', 'yield'])
# Extract the feature importances into a dataframe
feature_results = pd.DataFrame({'feature': list(train_features.columns), 
                                'importance': final_model.feature_importances_})

# Show the top 10 most important
feature_results = feature_results.sort_values('importance', ascending = False).reset_index(drop=True)

feature_results.head(10)
Out[475]:
feature importance
0 4 0.047800
1 62 0.023703
2 1144 0.022820
3 2393 0.022171
4 2297 0.020770
5 2226 0.019661
6 2267 0.018839
7 2 0.016693
8 2179 0.016467
9 2241 0.015411

We could use feature importance to perform feature selection; however, this is not an important process when using boosting trees algorithm as is for a linear model as linear regression. It might bring the actual model to perform even poorer.

I will extract the most 10 important features and train both a linear regression and a boosting tree algorithm and compare their performance.

In [476]:
# Extract the names of the most important features
most_important_features = feature_results['feature'][:10]

# Find the index that corresponds to each feature name
indices = [list(train.columns).index(x) for x in most_important_features]

# Keep only the most important features
x_train_reduced = x_train[:, indices]
x_valid_reduced = x_valid[:, indices]

print('Most important training features shape: ', x_train_reduced.shape)
print('Most important validation  features shape: ', x_valid_reduced.shape)
Most important training features shape:  (92, 10)
Most important validation  features shape:  (23, 10)
In [477]:
lr = LinearRegression()

# Fit on full set of features
lr.fit(x_train, y_train)
lr_full_pred = lr.predict(x_valid)

# Fit on reduced set of features
lr.fit(x_train_reduced, y_train)
lr_reduced_pred = lr.predict(x_valid_reduced)

# Display results
print('Linear Regression Full Results: MAE =    %0.4f.' % mape(y_valid, lr_full_pred))
print('Linear Regression Reduced Results: MAE = %0.4f.' % mape(y_valid, lr_reduced_pred))
Linear Regression Full Results: MAE =    16.1531.
Linear Regression Reduced Results: MAE = 15.2784.

Well, reducing the features did improve the linear regression results! It turns out that eliminating the features with low importance do actually improve performance in case of linear regression.

In [478]:
# Create the model with the same hyperparamters
model_reduced = GradientBoostingRegressor(alpha=0.9, criterion='friedman_mse', init=None,
             learning_rate=0.1, loss='ls', max_depth=2, max_features=None,
             max_leaf_nodes=None, min_impurity_decrease=0.0,
             min_impurity_split=None, min_samples_leaf=4,
             min_samples_split=6, min_weight_fraction_leaf=0.0,
             n_estimators=200, presort='auto', random_state=42,
             subsample=1.0, verbose=0, warm_start=False)

# Fit and test on the reduced set of features
model_reduced.fit(x_train_reduced, y_train)
model_reduced_pred = model_reduced.predict(x_valid_reduced)

print('Gradient Boosted Reduced Results: MAE = %0.4f' % mape(y_valid, model_reduced_pred))
Gradient Boosted Reduced Results: MAE = 15.7505

It seems that it does not happen the same thing for the gradient boosting algorithm, where the extra information in the features with low importance improve performance.

Locally Interpretable Model-agnostic Explanations

I will use LIME to explain individual predictions made by the model. LIME is aimed at showing how a machine learning model thinks by approximating the region around a prediction with a linear model.

I will look at trying to explain the predictions on an example the model gets very wrong and an example the model gets correct(well, slmodt correct!). I will restrict to using the reduced set of 10 features to aid interpretability. The model trained on the 10 most important features is much less accurate, but we generally have to trade off accuracy for interpretability!

In [479]:
# Find the residuals
y_valid= np.array(y_valid)
residuals = abs(model_reduced_pred - y_valid)
    
# Exact the worst and best prediction
wrong = x_valid_reduced[np.argmax(residuals), :]
right = x_valid_reduced[np.argmin(residuals), :]
In [480]:
# Create a lime explainer object
explainer = lime.lime_tabular.LimeTabularExplainer(training_data = x_train_reduced, 
                                                   mode = 'regression',
                                                   training_labels = y_train,
                                                   feature_names = list(most_important_features))
In [481]:
# Display the predicted and true value for the wrong instance
print('Prediction: %0.4f' % model_reduced.predict(wrong.reshape(1, -1)))
print('Actual Value: %0.4f' % y_valid[np.argmax(residuals)])

# Explanation for wrong prediction
wrong_exp = explainer.explain_instance(data_row = wrong, 
                                       predict_fn = model_reduced.predict)

# Plot the prediction explaination
wrong_exp.as_pyplot_figure();
plt.title('Explanation of Prediction', size = 28);
plt.xlabel('Effect on Prediction', size = 22);
Prediction: 0.8343
Actual Value: 0.5700
In [483]:
wrong_exp.show_in_notebook(show_predicted_value=False)

In this example, our gradient boosted model predicted a yield of 0.8343 and the actual value was 0.57.

The plot from LIME is showing us the contribution to the final prediction from each of the features for the example. We can see that variable 2 increased the prediction because it was under 0.68. The variable 2241 on the other hand, dicreased the prediction because it was equal tp 0.17.

We can interpret this as saying that our model thought the yield would be much higher than it actually was because the variable 4 was high. However, in this case, the yield was 0.57 despite the high value of the variable 4. While this significant mistake (off by 0.264!) might initially have been confusing, now we can see that in reality, the model was reasoning through the problem and just arrived at the incorrect value! A human going over the same process probably would have arrived at the same conclusion (if they had the patience to go through all the data).

Now we can go through the same process with a prediction the model got correct.

In [484]:
print('Prediction: %0.4f' % model_reduced.predict(right.reshape(1, -1)))
print('Actual Value: %0.4f' % y_valid[np.argmin(residuals)])

# Explanation for wrong prediction
right_exp = explainer.explain_instance(right, model_reduced.predict, num_features=10)
right_exp.as_pyplot_figure();
plt.title('Explanation of Prediction', size = 28);
plt.xlabel('Effect on Prediction', size = 22);
Prediction: 0.6194
Actual Value: 0.6200
In [485]:
right_exp.show_in_notebook(show_predicted_value=False)

The correct value for this case was 0.6194 which our gradient boosted model got almost right with 0.62 !

The plot from LIME again shows the contribution to the prediciton of each of feature variables for the example. For instance, because the variable 2 was more than -0.04, that contributed slightly significant to a higher estimate of the yield. Likewise, the variable 4 being equal to -0.08 also negatively contributed to the final prediction.

Observing break down plots like these allow us to get an idea of how the model makes a prediction. This is probably most valuable for cases where the model is off by a large amount as we can inspect the errors and perhaps engineer better features or adjust the hyperparameters of the model to improve predictions for next time. The examples where the model is off the most could also be interesting edge cases to look at manually.

Examining a Single Decision Tree

One of the best part about a tree-based ensemble is that we can look at any individual estimator. Although the final model is composed of 200 decision trees, and looking at a single one is not indicative of the entire model, it still allows us to see the general idea of how a decision tree works. From there, it is a natural extension to imagine hundreds of these trees building off the mistakes of previous trees to make a final prediction (this is a significant oversimplification of how gradient boosting regression works!)

I will first extract a tree from the forest and then save it using sklearn.tree.export_graphviz. This saves the tree as a .dot file which can be converted to a png using command line instructions Anaconda terminal.

In [486]:
# Extract a single tree
import pydotplus
single_tree = model_reduced.estimators_[105][0]

export_graphviz(single_tree, out_file = 'images/tree.dot',
                     rounded = True, 
                     feature_names = most_important_features,
                     filled = True)

single_tree
Out[486]:
DecisionTreeRegressor(criterion='friedman_mse', max_depth=2,
           max_features=None, max_leaf_nodes=None,
           min_impurity_decrease=0.0, min_impurity_split=None,
           min_samples_leaf=4, min_samples_split=6,
           min_weight_fraction_leaf=0.0, presort='auto',
           random_state=<mtrand.RandomState object at 0x00000288A70889D8>,
           splitter='best')
In [103]:
# Convert to a png from the command line
# This requires the graphviz visualization library (https://www.graphviz.org/)

#from command line transform from .dot tp .png
# dot -Tpng images/tree.dot -o images/tree.png

Now we can take a look at the tree and try to intrepret it's decisions! The best way to think of a decision tree is as a series of yes/no questions, like a flowchart. We start at the top, called the root, and move our way down the tree, with the direction of travel determined by the answer to each equation.

For instance, here the first question we ask is: is the variable 2257 less than or equal to -0.031? If the answer is yes, then we move to the left and ask the question: is the variable 2257 less than or equal to -0.061? If the answer to the first question was no, we move to the right and ask the question is the variable 169 less than or equal to -0.176?

We continue this iterative process until we reach the bottom of the tree and end up in a leaf node. Here, the value we predict corresponds to the value shown in the node (the values in this tree appear to be the actual predictions divided by 100).

Each node has four different pieces of information:

The question: based on this answer we move right or left to the next node a layer down in the tree The friedman_mse: a measure of the error for all of the examples in a given node The samples: number of examples in a node The value: the prediction of the target for all examples in a node

Conclusion

This is a small tree, where the model estimates the same value for each example in a node; however there is not a larger error because (this is my guessing) all of the examples have the same target value. However, with a bigger tree depth, the model would have a greater error.

In this case, and also to answer the question "What will make this problem set easier to solve?", the algorithm would most probably do better with a larger data set (i.e. imput from more weeks), and could also grow in depth (not too large, as too large tree algorithms can lead to overfitting).

As an alternative model, I would use a Bayesian regression model, for which I would use Markov Chain Monte Carlo and the data to find most likely parameters for the selected posterior distribution, and further use the posterior probability as the prior for applying Bayes Rule using additional hypothetical weather conditions data. I would set this model to give me interval predictions, as this is more correctly than just a single value.